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3 In mathematics, to find any term of a sequence, we use explicit formulas. In this article, we will be discussing the explicit formulas in detail along with real life applications and common mistakes made by students.

4 <h2>What are Explicit Formulas?</h2>

5 In a<a>sequence</a>to find any<a>term</a>without knowing the previous term, we use the<a>explicit formula</a>. It is a formula used to find the nth term of a sequence based on its position. Let’s learn the explicit formula for different types of sequences:

6 Type of SequenceExplicit FormulaExampleArithmetic Sequencean = a + (n - 1)d, where a is the first term and d is the<a>common difference</a>

7 For the sequence: 2, 4, 6, 8,… The fifth term, a5 = a + (n - 1)d Here, n = 5 a = 2 d = 2 a5 = 2 + (5 - 1) 2 = 2 + 4 × 2 = 2 + 8 = 10

8 Geometric Sequence an = arn-1, where a is the first term and ‘r’ is the common<a>ratio</a>For the sequence: 2, 6, 18,… The fifth term, a5 = arn - 1 Here, n = 5 a = 2 r = 3 a5 = 2 × 3(5 - 1) = 2 × 34 = 2 × 81 = 162

9 Harmonic Sequence an = 1/(a + (n - 1)d), where a is the first term and d is the common difference.For the sequence: 1/3, 1/7, 1/11,…

10 <h2>How To Find Explicit Formulas?</h2>

11 To find the nth term of a sequence, we use an explicit<a>formula</a>. The sequence can be<a>arithmetic</a>, geometric, or harmonic.

12 nth term of an<a>arithmetic sequence</a>- an = a + (n - 1)d, where a is the first term

13 n is the position of the term in the sequence,

14 and d is the common difference. Steps used for finding explicit formulas are -

15 Step 1:First find the first term and the common difference of the sequence

16 Step 2:Substitute the value of a, n, and d in the explicit formula, an = a + (n - 1)d

17 Step 3:Simplify the formula to find the nth term

18 To find the 7th term of the sequence 7, 14, 21, 28, …

19 In the given sequence, a = 7 and d = 7(14 - 7 = 7)

20 The nth term of an arithmetic sequence is: an = a + (n - 1)d

21 So, the 7th term is: a7 = 7 + (7 - 1)7

22 = 7+ (6 × 7)

23 = 7 + 42 = 49

24 Therefore, the 7th term of the sequence is 49.

25 <h2>Explicit Formula For Arithmetic Sequence</h2>

26 In an arithmetic sequence, the difference between consecutive terms is<a>constant</a>and is called the common difference (d). To find the nth term of an arithmetic sequence, we use the explicit formula: an = a + (n - 1)d. Where,

27 <ul><li>an is the nth term</li>

28 </ul><ul><li>a is the first term</li>

29 </ul><ul><li>d is the common difference</li>

30 </ul>For example, for the arithmetic sequence 2, 5, 8, 11, 14, …, finding the explicit formula

31 Here, a = 2

32 d = 3

33 The explicit formula of an arithmetic sequence is: an = a + (n - 1)d Substituting the value of a and d:

34 an = 2 + (n - 1)3

35 = 2 + 3n - 3

36 an = 3n - 1

37 Find the 25th term of the sequence.

38 a25 = 3 × 25 - 1

39 = 75 - 1

40 = 74

41 Therefore, the 25th term of the sequence is 74.

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44 <h2>Explicit Formula For Geometric Sequence</h2>

43 <h2>Explicit Formula For Geometric Sequence</h2>

45 The<a>geometric sequence</a>is any sequence where the ratio of any two consecutive terms is the same. The ratio is known as the common ratio (r). The general form of a geometric sequence can be represented as a, ar, ar2, ar3, … arn - 1. For the geometric sequence, the explicit formula is an = arn - 1.

44 The<a>geometric sequence</a>is any sequence where the ratio of any two consecutive terms is the same. The ratio is known as the common ratio (r). The general form of a geometric sequence can be represented as a, ar, ar2, ar3, … arn - 1. For the geometric sequence, the explicit formula is an = arn - 1.

46 Where,

45 Where,

47 <ul><li>a is the first term</li>

46 <ul><li>a is the first term</li>

48 </ul><ul><li>an is the nth term </li>

47 </ul><ul><li>an is the nth term </li>

49 </ul><ul><li>r is the common ratio</li>

48 </ul><ul><li>r is the common ratio</li>

50 </ul>For example, for the sequence: 1, 2, 4, 8, …, finding the explicit formula

49 </ul>For example, for the sequence: 1, 2, 4, 8, …, finding the explicit formula

51 Here, a = 1

50 Here, a = 1

52 r = 2

51 r = 2

53 The explicit formula for geometric sequences is: an = arn - 1 Substituting the value of a and r:

52 The explicit formula for geometric sequences is: an = arn - 1 Substituting the value of a and r:

54 an = 1 × 2n - 1 = 2n - 1

53 an = 1 × 2n - 1 = 2n - 1

55 Finding the 5th term:

54 Finding the 5th term:

56 a5 = 2(5 - 1)

55 a5 = 2(5 - 1)

57 = 24 = 16

56 = 24 = 16

58 So, the 5th term is 16

57 So, the 5th term is 16

59 <h2>Explicit Formula For Harmonic Sequence</h2>

58 <h2>Explicit Formula For Harmonic Sequence</h2>

60 A harmonic sequence is a type of sequence where the reciprocals of the terms form an arithmetic sequence. For instance, the harmonic sequence of 2, 4, 6, 8, … is 1/2, 1/4, 1/6, 1/8, …. The general form of a harmonic sequence can be represented as 1/a, 1/(a +d), 1/(a + 2d), …, 1/(a + (n - 1)d). For a harmonic sequence, the explicit formula: an = 1/(a + (n - 1)d)

59 A harmonic sequence is a type of sequence where the reciprocals of the terms form an arithmetic sequence. For instance, the harmonic sequence of 2, 4, 6, 8, … is 1/2, 1/4, 1/6, 1/8, …. The general form of a harmonic sequence can be represented as 1/a, 1/(a +d), 1/(a + 2d), …, 1/(a + (n - 1)d). For a harmonic sequence, the explicit formula: an = 1/(a + (n - 1)d)

61 <ul><li>Where a is the first term</li>

60 <ul><li>Where a is the first term</li>

62 </ul><ul><li>an is the nth term</li>

61 </ul><ul><li>an is the nth term</li>

63 </ul><ul><li>d is the common difference</li>

62 </ul><ul><li>d is the common difference</li>

64 </ul>For example, find the explicit formula for the harmonic sequence: 1/3, 1/6, 1/9, 1/12.

63 </ul>For example, find the explicit formula for the harmonic sequence: 1/3, 1/6, 1/9, 1/12.

65 We take the reciprocals of the terms: 3, 6, 9, 12, …, to find a and d. Here, a = 3

64 We take the reciprocals of the terms: 3, 6, 9, 12, …, to find a and d. Here, a = 3

66 d = 3

65 d = 3

67 an = 1/(a + (n - 1)d)

66 an = 1/(a + (n - 1)d)

68 Substituting the value of a and d

67 Substituting the value of a and d

69 an = 1/(3 + (n - 1)3)

68 an = 1/(3 + (n - 1)3)

70 = 1/(3 + 3n - 3)

69 = 1/(3 + 3n - 3)

71 = 1/(3n)

70 = 1/(3n)

72 Finding the 5th term

71 Finding the 5th term

73 an = 1/(3n)

72 an = 1/(3n)

74 a5 = 1/(3 × 5)

73 a5 = 1/(3 × 5)

75 = 1/15

74 = 1/15

76 So, the 5th term of the sequence is 1/15

75 So, the 5th term of the sequence is 1/15

77 <h2>Step-by-Step Use of an Explicit Formula</h2>

76 <h2>Step-by-Step Use of an Explicit Formula</h2>

78 The use of explicit formula can be understood by the example mentioned below.

77 The use of explicit formula can be understood by the example mentioned below.

79 Find the 5th term of the sequence where an=3n+2.

78 Find the 5th term of the sequence where an=3n+2.

80 Step 1:Identify the formula and substitute n = 5n

79 Step 1:Identify the formula and substitute n = 5n

81 a5 = 3(5)+2

80 a5 = 3(5)+2

82 Step 2:Simplify the<a>expression</a>.

81 Step 2:Simplify the<a>expression</a>.

83 a5 = 15 + 2 = 17.

82 a5 = 15 + 2 = 17.

84 The fifth term of the sequence is 17.

83 The fifth term of the sequence is 17.

85 <h2>Tips and Tricks to Master Explicit Formulas</h2>

84 <h2>Tips and Tricks to Master Explicit Formulas</h2>

86 Explicit formulas in<a>math</a>are a difficult topic to comprehend, therefore some tips and tricks are useful to master the topic.

85 Explicit formulas in<a>math</a>are a difficult topic to comprehend, therefore some tips and tricks are useful to master the topic.

87 Always Identify the Type of Sequence First: Before using any formula, check if it’s arithmetic, geometric, or harmonic.

86 Always Identify the Type of Sequence First: Before using any formula, check if it’s arithmetic, geometric, or harmonic.

88 Write Down the Given Data Clearly: Always list what is given, a1, n, d, or r.

87 Write Down the Given Data Clearly: Always list what is given, a1, n, d, or r.

89 Substitute Carefully: When substituting into the formula, use parentheses to avoid sign or order errors.

88 Substitute Carefully: When substituting into the formula, use parentheses to avoid sign or order errors.

90 Don’t Forget the Order of Operations: Follow BODMAS/PEDMAS rules while simplifying.

89 Don’t Forget the Order of Operations: Follow BODMAS/PEDMAS rules while simplifying.

91 Watch for Negative or Fractional Values: If r<0 or d<0 o, terms may alternate signs.

90 Watch for Negative or Fractional Values: If r<0 or d<0 o, terms may alternate signs.

92 <h2>Common Mistakes and How to Avoid Them in Explicit Formulas</h2>

91 <h2>Common Mistakes and How to Avoid Them in Explicit Formulas</h2>

93 The explicit formula is used to find the nth term of a sequence. Students tend to make mistakes when using the explicit formula. Here are some common mistakes and the ways to avoid them.

92 The explicit formula is used to find the nth term of a sequence. Students tend to make mistakes when using the explicit formula. Here are some common mistakes and the ways to avoid them.

94 <h2>Real-world Applications of Explicit Formulas</h2>

93 <h2>Real-world Applications of Explicit Formulas</h2>

95 To calculate or predict a specific term in a sequence, we use the explicit formulas. Here are some real-life applications of the explicit formulas.

94 To calculate or predict a specific term in a sequence, we use the explicit formulas. Here are some real-life applications of the explicit formulas.

96 <ul><li>In sports, we use the arithmetic sequence to design a workout routine to plan a steady increase in the repetitions. At any point in training to determine the<a>number</a>of repetitions, we use the explicit formula. </li>

95 <ul><li>In sports, we use the arithmetic sequence to design a workout routine to plan a steady increase in the repetitions. At any point in training to determine the<a>number</a>of repetitions, we use the explicit formula. </li>

97 </ul><ul><li>In finance, to calculate the savings, loans, and investments, where the amount forms a geometric sequence, we can use the explicit formula to predict the amount at a particular time. </li>

96 </ul><ul><li>In finance, to calculate the savings, loans, and investments, where the amount forms a geometric sequence, we can use the explicit formula to predict the amount at a particular time. </li>

98 </ul><ul><li>For linear population growth models, the explicit formula can predict the population at a specific time. </li>

97 </ul><ul><li>For linear population growth models, the explicit formula can predict the population at a specific time. </li>

99 </ul><ul><li>In computer algorithms, we can use explicit formulas to directly calculate the value in the sequence, which helps in improving efficiency. </li>

98 </ul><ul><li>In computer algorithms, we can use explicit formulas to directly calculate the value in the sequence, which helps in improving efficiency. </li>

100 </ul><ul><li>In environmental studies, explicit formulas can be applied to model patterns such as temperature changes or rainfall distribution over time, helping predict future trends based on previous<a>data</a>.</li>

99 </ul><ul><li>In environmental studies, explicit formulas can be applied to model patterns such as temperature changes or rainfall distribution over time, helping predict future trends based on previous<a>data</a>.</li>

101 </ul><h3>Problem 1</h3>

100 </ul><h3>Problem 1</h3>

102 Find the explicit formula for an arithmetic sequence, where the first term is 5 and the common difference is 3.

101 Find the explicit formula for an arithmetic sequence, where the first term is 5 and the common difference is 3.

103 Okay, lets begin

102 Okay, lets begin

104 an = 3n + 2

103 an = 3n + 2

105 <h3>Explanation</h3>

104 <h3>Explanation</h3>

106 To find the explicit formula of an arithmetic sequence, we use the formula, an = a + (n - 1)d

105 To find the explicit formula of an arithmetic sequence, we use the formula, an = a + (n - 1)d

107 Here, a = 5

106 Here, a = 5

108 d = 3

107 d = 3

109 Therefore, an = 5 + (n - 1)3

108 Therefore, an = 5 + (n - 1)3

110 = 5 + 3n - 3

109 = 5 + 3n - 3

111 = 3n + 2

110 = 3n + 2

112 Well explained 👍

111 Well explained 👍

113 <h3>Problem 2</h3>

112 <h3>Problem 2</h3>

114 If the first term and the common ratio of a geometric sequence are 3 and 2, find the explicit formula.

113 If the first term and the common ratio of a geometric sequence are 3 and 2, find the explicit formula.

115 Okay, lets begin

114 Okay, lets begin

116 an = 3 × 2n -1

115 an = 3 × 2n -1

117 <h3>Explanation</h3>

116 <h3>Explanation</h3>

118 The explicit formula of a geometric sequence is: an = arn - 1

117 The explicit formula of a geometric sequence is: an = arn - 1

119 Here, a = 3

118 Here, a = 3

120 r = 2

119 r = 2

121 So, an = 3 × 2n - 1

120 So, an = 3 × 2n - 1

122 Well explained 👍

121 Well explained 👍

123 <h3>Problem 3</h3>

122 <h3>Problem 3</h3>

124 Find the 25th term of a harmonic sequence where the first term is 1/2 and the common difference is 3?

123 Find the 25th term of a harmonic sequence where the first term is 1/2 and the common difference is 3?

125 Okay, lets begin

124 Okay, lets begin

126 The 25th term is 1/74

125 The 25th term is 1/74

127 <h3>Explanation</h3>

126 <h3>Explanation</h3>

128 The explicit formula for a harmonic sequence is: an = 1/(a + (n - 1)d)

127 The explicit formula for a harmonic sequence is: an = 1/(a + (n - 1)d)

129 Given the first term is 1/2

128 Given the first term is 1/2

130 So, a = 2

129 So, a = 2

131 d = 3

130 d = 3

132 So, an = 1/(2 + (n - 1)3)

131 So, an = 1/(2 + (n - 1)3)

133 = 1/(2 + 3n - 3)

132 = 1/(2 + 3n - 3)

134 = 1/(3n -1)

133 = 1/(3n -1)

135 So, the 25th term = a25 = 1/(3 × 25 - 1)

134 So, the 25th term = a25 = 1/(3 × 25 - 1)

136 = 1/(75 - 1)

135 = 1/(75 - 1)

137 = 1/74

136 = 1/74

138 So, the 25th term is 1/74

137 So, the 25th term is 1/74

139 Well explained 👍

138 Well explained 👍

140 <h3>Problem 4</h3>

139 <h3>Problem 4</h3>

141 For an arithmetic sequence with a = 3 and d = 5, find the 12th term.

140 For an arithmetic sequence with a = 3 and d = 5, find the 12th term.

142 Okay, lets begin

141 Okay, lets begin

143 The 12th term is 58

142 The 12th term is 58

144 <h3>Explanation</h3>

143 <h3>Explanation</h3>

145 If a = 3 and d = 5

144 If a = 3 and d = 5

146 The explicit formula of the arithmetic sequence is: an = a + (n - 1)d

145 The explicit formula of the arithmetic sequence is: an = a + (n - 1)d

147 a12 = 3 + (12 - 1)5

146 a12 = 3 + (12 - 1)5

148 = 3 + 11 × 5

147 = 3 + 11 × 5

149 = 3 + 55

148 = 3 + 55

150 = 58

149 = 58

151 So, here the 12th term is 58

150 So, here the 12th term is 58

152 Well explained 👍

151 Well explained 👍

153 <h3>Problem 5</h3>

152 <h3>Problem 5</h3>

154 What is the common difference of the sequence if the explicit formula is a_n = 7n - 2?

153 What is the common difference of the sequence if the explicit formula is a_n = 7n - 2?

155 Okay, lets begin

154 Okay, lets begin

156 The common difference is 7

155 The common difference is 7

157 <h3>Explanation</h3>

156 <h3>Explanation</h3>

158 Here, the explicit formula is: an = 7n - 2

157 Here, the explicit formula is: an = 7n - 2

159 To find the common difference, let’s find the 2nd and 1st terms

158 To find the common difference, let’s find the 2nd and 1st terms

160 a1 = 7 × 1 - 2

159 a1 = 7 × 1 - 2

161 = 7 - 2

160 = 7 - 2

162 = 5

161 = 5

163 a2 = 7 × 2 - 2

162 a2 = 7 × 2 - 2

164 = 14 - 2

163 = 14 - 2

165 = 12

164 = 12

166 The difference between any two consecutive terms in a sequence is the common difference.

165 The difference between any two consecutive terms in a sequence is the common difference.

167 So, d = a2 - a1

166 So, d = a2 - a1

168 = 12 - 5

167 = 12 - 5

169 = 7

168 = 7

170 Well explained 👍

169 Well explained 👍

171 <h2>FAQs on Explicit Formulas</h2>

170 <h2>FAQs on Explicit Formulas</h2>

172 <h3>1.What is an explicit formula?</h3>

171 <h3>1.What is an explicit formula?</h3>

173 The formula used to find the nth term of a sequence when the previous term is unknown is the explicit formula.

172 The formula used to find the nth term of a sequence when the previous term is unknown is the explicit formula.

174 <h3>2.What is the explicit formula for an arithmetic sequence?</h3>

173 <h3>2.What is the explicit formula for an arithmetic sequence?</h3>

175 For an arithmetic sequence, the explicit formula is an = a + (n - 1)d, where a is the first term and d is the common difference.

174 For an arithmetic sequence, the explicit formula is an = a + (n - 1)d, where a is the first term and d is the common difference.

176 <h3>3.What is the explicit formula for a geometric sequence?</h3>

175 <h3>3.What is the explicit formula for a geometric sequence?</h3>

177 The explicit formula for geometric sequences is an = arn - 1, where a is the first term and r is the common difference.

176 The explicit formula for geometric sequences is an = arn - 1, where a is the first term and r is the common difference.

178 <h3>4.What is the explicit formula for a harmonic sequence?</h3>

177 <h3>4.What is the explicit formula for a harmonic sequence?</h3>

179 The formula for harmonic sequence is: an = 1/(a + (n - 1)d).

178 The formula for harmonic sequence is: an = 1/(a + (n - 1)d).

180 <h3>5.What is the explicit formula for 2, 4, 6, 8, ….?</h3>

179 <h3>5.What is the explicit formula for 2, 4, 6, 8, ….?</h3>

181 an = a + (n - 1)d is the explicit formula for the arithmetic sequence. Here a = 2 and d = 2, so an = 2 + (n - 1)2 = 2 + 2n - 2, so an = 2n.

180 an = a + (n - 1)d is the explicit formula for the arithmetic sequence. Here a = 2 and d = 2, so an = 2 + (n - 1)2 = 2 + 2n - 2, so an = 2n.

182 <h2>Jaskaran Singh Saluja</h2>

181 <h2>Jaskaran Singh Saluja</h2>

183 <h3>About the Author</h3>

182 <h3>About the Author</h3>

184 Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

183 Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

185 <h3>Fun Fact</h3>

184 <h3>Fun Fact</h3>

186 : He loves to play the quiz with kids through algebra to make kids love it.

185 : He loves to play the quiz with kids through algebra to make kids love it.