Sequences and Series — Previous Year Questions (Class 11 Mathematics)
Sequences and Series involve patterns of numbers. Learn arithmetic, geometric, and special sequences to solve problems about sums and patterns.
TL;DR: Sequences and Series involve patterns of numbers. Learn arithmetic, geometric, and special sequences to solve problems about sums and patterns.
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Sequences and Series involve patterns of numbers. Learn arithmetic, geometric, and special sequences to solve problems about sums and patterns.
Sequences and Series — Previous Year Questions with Solutions
Q (2023, 1 mark): Find the 10th term of the arithmetic sequence 2, 5, 8, 11, ...
Answer: Given AP: 2, 5, 8, 11, ...
First term (a) = 2
Common difference (d) = 5 - 2 = 3
n = 10
Using the formula for nth term:
a_n = a + (n-1)d
a_10 = 2 + (10-1)(3)
a_10 = 2 + 9(3)
a_10 = 2 + 27
a_10 = 29
10th term = 29
Q (2022, 2 marks): Find the sum of the first 20 terms of the arithmetic sequence 3, 7, 11, 15, ...
Answer: Given AP: 3, 7, 11, 15, ...
First term (a) = 3
Common difference (d) = 7 - 3 = 4
n = 20
Using the formula for sum of n terms:
S_n = n/2[2a + (n-1)d]
S_20 = 20/2[2(3) + (20-1)(4)]
S_20 = 10[6 + 19(4)]
S_20 = 10[6 + 76]
S_20 = 10[82]
S_20 = 820
Sum of first 20 terms = 820
Q (2024, 2 marks): Find the 8th term of the geometric sequence 1, 3, 9, 27, ...
Answer: Given GP: 1, 3, 9, 27, ...
First term (a) = 1
Common ratio (r) = 3/1 = 3
n = 8
Using the formula for nth term:
a_n = a × r^(n-1)
a_8 = 1 × 3^(8-1)
a_8 = 1 × 3^7
a_8 = 2187
8th term = 2187
Q (2023, 3 marks): Find the sum of the first 6 terms of the geometric sequence 2, 6, 18, 54, ...
Answer: Given GP: 2, 6, 18, 54, ...
First term (a) = 2
Common ratio (r) = 6/2 = 3
n = 6
Using the formula for sum of n terms (when r ≠ 1):
S_n = a(r^n - 1)/(r - 1)
S_6 = 2(3^6 - 1)/(3 - 1)
S_6 = 2(729 - 1)/2
S_6 = 2(728)/2
S_6 = 728
Sum of first 6 terms = 728
Q (2022, 3 marks): Which term of the AP 5, 12, 19, 26, ... is 89?
Answer: Given AP: 5, 12, 19, 26, ...
First term (a) = 5
Common difference (d) = 12 - 5 = 7
Required term = 89
Using the formula:
a_n = a + (n-1)d
89 = 5 + (n-1)(7)
89 - 5 = (n-1)(7)
84 = (n-1)(7)
n - 1 = 84/7
n - 1 = 12
n = 13
89 is the 13th term
Q (2024, 2 marks): Find the sum of the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ...
Answer: Given infinite GP: 1 + 1/2 + 1/4 + 1/8 + ...
First term (a) = 1
Common ratio (r) = (1/2)/1 = 1/2
Since |r| < 1, the series converges.
Using the formula for infinite geometric series:
S_∞ = a/(1 - r)
S_∞ = 1/(1 - 1/2)
S_∞ = 1/(1/2)
S_∞ = 2
Sum of infinite series = 2
Frequently Asked Questions
What is the difference between an arithmetic sequence and a geometric sequence?
An arithmetic sequence has a constant difference (d) between consecutive terms, while a geometric sequence has a constant ratio (r) between consecutive terms. In an AP, each term is found by adding d to the previous term. In a GP, each term is found by multiplying the previous term by r.
When does an infinite geometric series converge?
An infinite geometric series converges when the absolute value of the common ratio is less than 1, i.e., |r| < 1. If |r| ≥ 1, the series diverges. When it converges, the sum is S = a/(1-r).
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