Sequences, Series, and Probability — High School Pre-Calculus practice questions

What is the nth term of an arithmetic sequence with first term a₁ and common difference d?

1. aₙ = a₁ + nd
2. aₙ = a₁ · dⁿ⁻¹
3. aₙ = a₁ + (n − 1)d
4. aₙ = nd

Answer: C — aₙ = a₁ + (n − 1)d

C) Correct — There are n − 1 steps of d from the first term. A) Counts one step too many. B) Is the geometric formula. D) Leaves out the first term.

a₁ = 5 and aₙ = aₙ₋₁ + 3. Find a₅.

1. 20
2. 17
3. 15
4. 14

Answer: B — 17

B) Correct — 5, 8, 11, 14, 17. A) Adds 3 five times instead of four. C) Multiplies 5 × 3. D) Is a₄.

Find the sum of the infinite geometric series with first term 4 and ratio 1/3.

1. 12
2. 4/3
3. 6
4. It diverges

Answer: C — 6

C) Correct — S = a/(1 − r) = 4/(2/3). A) Divides a by r. B) Multiplies a by r. D) |r| < 1, so it converges.

How many ways can 2 people be chosen from 6, when order does not matter?

1. 15
2. 30
3. 12
4. 36

Answer: A — 15

A) Correct — C(6, 2) = 6 · 5/2. B) Counts ordered pairs, P(6, 2). C) Multiplies 6 × 2. D) Squares 6.

Find 1 + 2 + 3 + … + 100.

1. 5000
2. 10,100
3. 10,000
4. 5050

Answer: D — 5050

D) Correct — n(n + 1)/2 = 100 · 101/2. A) Multiplies 100 by 50, missing the extra 50. B) Forgets to divide by 2. C) Squares 100.

An infinite geometric series converges when:

1. r > 0
2. r < 1
3. Always
4. |r| < 1

Answer: D — |r| < 1

D) Correct — Only then do the terms shrink toward 0 fast enough. A) r = 2 is positive but diverges. B) Misses that r ≤ −1 diverges too. C) r = 2 gives 1 + 2 + 4 + …, which diverges.

What is the coefficient of x²y³ in (x + y)⁵?

1. 5
2. 10
3. 6
4. 20

Answer: B — 10

B) Correct — C(5, 3) = 10. A) Uses n instead of C(n, k). C) Adds the exponents and 1. D) Multiplies 5 × 4 and forgets to divide by 2.

What is the difference between a permutation and a combination?

Permutation: order matters (P(n,k) = n!/(n−k)!). Combination: order does not (C(n,k) = n!/(k!(n−k)!)).

What is the formula for the sum of the first n terms of a geometric series?

Sₙ = a₁(1 − rⁿ)/(1 − r), for r ≠ 1.

What is C(n, k)?

n! / (k!(n−k)!) — the number of ways to choose k items from n.

Worked Example: Find the sum of the infinite geometric series 8 + 4 + 2 + 1 + ...

Step 1: Identify the first term and common ratio. a = 8 r = 4/8 = 1/2 Step 2: Check convergence. |r| = 1/2 < 1 → the series converges. Step 3: Apply the formula S = a/(1 − r). S = 8/(1 − 1/2) = 8/(1/2) = 16 Step 4 (sanity check): Partial sums approach 16: 8, 12, 14, 15, 15.5, 15.75, ... → 16. Final answer: S = 16.