📘 Precalculus Sequences Series Telescoping
Recommended: 30–40 minutes. No calculator.
Problems
1.
Tags: Arithmetic Sequence · Easy · source: Original (AMC-style)
What is the 5th term of the sequence 2, 5, 8, 11, …?
A) $14$ B) $17$ C) $20$ D) $23$ E) $26$
Answer & Solution
Answer: A
This is an arithmetic sequence with first term 2 and common difference 3. The 5th term is 2 + 4(3) = 14.
2.
Tags: Geometric Sequence · Easy · source: Original (AMC-style)
What is the 4th term of the sequence 3, 6, 12, …?
A) $18$ B) $24$ C) $30$ D) $36$ E) $42$
Answer & Solution
Answer: B
This is a geometric sequence with first term 3 and common ratio 2. The 4th term is 3 × 2³ = 3 × 8 = 24.
3.
Tags: Arithmetic Sum · Easy · source: Original (AMC-style)
What is the sum of the first 5 positive integers?
A) $10$ B) $12$ C) $15$ D) $18$ E) $20$
Answer & Solution
Answer: C
Sum = 1 + 2 + 3 + 4 + 5 = 15.
4.
Tags: Geometric Sum · Easy · source: Original (AMC-style)
What is the sum of 1 + 2 + 4 + 8?
A) $12$ B) $15$ C) $16$ D) $18$ E) $20$
Answer & Solution
Answer: B
Sum = 1 + 2 + 4 + 8 = 15.
5.
Tags: Fibonacci · Easy · source: Original (AMC-style)
What is the 6th Fibonacci number?
A) $5$ B) $8$ C) $13$ D) $21$ E) $34$
Answer & Solution
Answer: B
Fibonacci sequence: 1, 1, 2, 3, 5, 8, ... The 6th term is 8.
6.
Tags: Sequence Pattern · Easy · source: Original (AMC-style)
What is the next term in 1, 4, 9, 16, …?
A) $20$ B) $24$ C) $25$ D) $30$ E) $36$
Answer & Solution
Answer: C
This is the sequence of perfect squares: 1², 2², 3², 4², 5² = 25.
7.
Tags: Arithmetic Formula · Medium · source: Original (AMC-style)
What is the sum of the first 10 positive integers?
A) $45$ B) $50$ C) $55$ D) $60$ E) $65$
Answer & Solution
Answer: C
Using the formula: Sum = n(n+1)/2 = 10(11)/2 = 55.
8.
Tags: Geometric Formula · Medium · source: Original (AMC-style)
What is the sum of 1 + 3 + 9 + 27 + 81?
A) $100$ B) $121$ C) $130$ D) $140$ E) $150$
Answer & Solution
Answer: B
This is a geometric series with first term 1 and ratio 3. Sum = 1(3⁵-1)/(3-1) = (243-1)/2 = 121.
9.
Tags: Telescoping · Medium · source: Original (AMC-style)
What is the sum of 1/2 + 1/6 + 1/12 + 1/20?
A) $1/4$ B) $1/3$ C) $1/2$ D) $2/3$ E) $3/4$
Answer & Solution
Answer: E
Using telescoping: 1/2 + 1/6 + 1/12 + 1/20 = (1-1/2) + (1/2-1/3) + (1/3-1/4) + (1/4-1/5) = 1 - 1/5 = 4/5.
10.
Tags: Infinite Series · Medium · source: Original (AMC-style)
What is the sum of 1 + 1/2 + 1/4 + 1/8 + …?
A) $1$ B) $2$ C) $3$ D) $4$ E) Infinite
Answer & Solution
Answer: B
This is an infinite geometric series with first term 1 and ratio 1/2. Sum = 1/(1-1/2) = 1/(1/2) = 2.
11.
Tags: Advanced Telescoping · Hard · source: Original (AMC-style)
What is the sum of 1/(1×2) + 1/(2×3) + 1/(3×4) + … + 1/(9×10)?
A) $1/10$ B) $1/9$ C) $9/10$ D) $1$ E) $10/9$
Answer & Solution
Answer: C
Using telescoping: 1/(1×2) + 1/(2×3) + ... + 1/(9×10) = (1-1/2) + (1/2-1/3) + ... + (1/9-1/10) = 1 - 1/10 = 9/10.
12.
Tags: Complex Series · Hard · source: Original (AMC-style)
What is the sum of 1 + 2 + 3 + … + 100?
A) $4950$ B) $5000$ C) $5050$ D) $5100$ E) $5150$
Answer & Solution
Answer: C
Using the formula: Sum = n(n+1)/2 = 100(101)/2 = 5050.
Answer Key
| # | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Ans | A | B | C | B | B | C | C | B | E | B | C | C |