Sequences and Series Solved Examples (11 Mathematics)
Sequences are ordered lists of numbers; series are sums of sequence terms. These examples cover arithmetic and geometric sequences, their sums, and applica
TL;DR: Sequences are ordered lists of numbers; series are sums of sequence terms. These examples cover arithmetic and geometric sequences, their sums, and ap…
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Sequences are ordered lists of numbers; series are sums of sequence terms. These examples cover arithmetic and geometric sequences, their sums, and applica
Sequences and Series — Solved Numerical Examples (Step by Step)
Example 1: Find the 10th term of the arithmetic sequence 5, 9, 13, 17, ...
Solution: For arithmetic sequence with first term a and common difference d:
a_n = a + (n-1)d
Given: a = 5, d = 9 - 5 = 4, n = 10
a_10 = 5 + (10-1) × 4
a_10 = 5 + 9 × 4
a_10 = 5 + 36 = 41
Example 2: Find the sum of first 20 terms of the arithmetic sequence 2, 5, 8, 11, ...
Solution: For arithmetic sequence:
S_n = n/2 × (2a + (n-1)d) = n/2 × (a + l)
Where l is the last term
Given: a = 2, d = 3, n = 20
Method 1: S_20 = 20/2 × (2(2) + (20-1) × 3)
S_20 = 10 × (4 + 57)
S_20 = 10 × 61 = 610
Method 2: First find 20th term:
a_20 = 2 + 19(3) = 2 + 57 = 59
S_20 = 20/2 × (2 + 59) = 10 × 61 = 610
Example 3: Find the 5th term of the geometric sequence 2, 6, 18, 54, ...
Solution: For geometric sequence:
a_n = a × r^(n-1)
Where r is common ratio
Given: a = 2, r = 6/2 = 3, n = 5
a_5 = 2 × 3^(5-1)
a_5 = 2 × 3^4
a_5 = 2 × 81 = 162
Example 4: Find the sum of first 8 terms of the geometric sequence 1, 2, 4, 8, ...
Solution: For geometric sequence:
S_n = a(r^n - 1) / (r - 1) when r ≠ 1
Given: a = 1, r = 2, n = 8
S_8 = 1 × (2^8 - 1) / (2 - 1)
S_8 = (256 - 1) / 1
S_8 = 255
Example 5: Find the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ...
Solution: For infinite geometric series with |r| < 1:
S_∞ = a / (1 - r)
Given: a = 1, r = 1/2
Since |r| = 1/2 < 1, series converges.
S_∞ = 1 / (1 - 1/2)
S_∞ = 1 / (1/2) = 2
Example 6: How many terms of the arithmetic sequence 5, 8, 11, ... sum to 185?
Solution: Given: a = 5, d = 3, S_n = 185
Find: n
Using S_n = n/2 × (2a + (n-1)d)
185 = n/2 × (2(5) + (n-1) × 3)
185 = n/2 × (10 + 3n - 3)
185 = n/2 × (7 + 3n)
370 = n(7 + 3n)
370 = 7n + 3n²
3n² + 7n - 370 = 0
Using quadratic formula:
n = (-7 ± √(49 + 4440)) / 6 = (-7 ± √4489) / 6
n = (-7 ± 67) / 6
n = 60/6 = 10 or n = -74/6 (rejected, negative)
n = 10
Example 7: Find the sum of 1² + 2² + 3² + ... + 10².
Solution: Use formula for sum of squares:
Σ(k²) from k=1 to n = n(n+1)(2n+1) / 6
For n = 10:
Σ(k²) = 10 × 11 × 21 / 6
Σ(k²) = 2310 / 6
Σ(k²) = 385
Tips
- Arithmetic sequence: constant difference d between consecutive terms; a_n = a + (n-1)d.
- Geometric sequence: constant ratio r between consecutive terms; a_n = a × r^(n-1).
- Sum formulas: S_n = n/2(first + last) for AP; S_n = a(r^n - 1)/(r - 1) for GP.
- Infinite geometric series converges only when |r| < 1; sum = a/(1-r).
Frequently Asked Questions
When does an infinite geometric series converge?
An infinite geometric series converges only when the absolute value of the common ratio is less than 1 (|r| < 1). If |r| >= 1, the series diverges (sum is infinite or undefined).
What is the difference between a sequence and a series?
A sequence is a list of numbers in a specific order. A series is the sum of the terms of a sequence. For example, 2, 4, 6, 8 is a sequence; 2 + 4 + 6 + 8 = 20 is a series.
More Mathematics Solved Examples
- Real Numbers (HCF LCM Euclid)
- Pair of Linear Equations in Two Variables
- Circles
- Statistics
- Linear Equations in Two Variables
- Heron's Formula
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