The partial sums of the geometric sequence 1, 1/2, 1/4, 1/8, ... are computed up to 10 terms, then 100 terms, then 1000 terms. What happens to these partial sums?
AThey grow without bound, since we are always adding positive terms
BThey approach 2, since the infinite geometric series converges to a₁/(1-r) = 1/(1-½) = 2
CThey approach 1, since the first term is 1 and the remaining terms are increasingly negligible
DThey fluctuate, since the terms alternate between larger and smaller values
The series has first term a₁ = 1 and ratio r = 1/2. Since |r| < 1, the partial sum formula Sₙ = a₁(1-rⁿ)/(1-r) = 2(1-(1/2)ⁿ) approaches 2 as n→∞. Option C is the tempting trap: students assume the first term dominates, but the infinite tail sums contribute a full additional 1 to the total. Adding infinitely many positive terms can still yield a finite sum — this is the key insight that launches all of Calculus 2 series convergence theory.
Question 2 Multiple Choice
You are given the sequence 3, 6, 12, 24, 48, ... and asked to find the sum of the first 7 terms. Which formula applies?
ASₙ = n/2 · (a₁ + aₙ), because the sequence has a constant second difference
BSₙ = a₁(1-rⁿ)/(1-r) with r = 2, because this is a geometric sequence
CSₙ = a₁(1-rⁿ)/(1-r) with r = 3, because each term adds 3
DSₙ = n/2 · (a₁ + aₙ), because each term is double the previous
Each term is multiplied by 2 (constant ratio r = 2), making this a geometric sequence. The geometric partial sum formula gives S₇ = 3(1-2⁷)/(1-2) = 3(-127)/(-1) = 381. The arithmetic sum formula (average of first and last, times n) applies only when there is a constant difference. Identifying the sequence type before selecting a formula is the essential first step — applying the arithmetic formula here would give the wrong answer.
Question 3 True / False
The series 1 + 3 + 5 + 7 + ... (the sum of odd numbers) and the sequence 1, 3, 5, 7, ... are two names for the same mathematical object.
TTrue
FFalse
Answer: False
The sequence 1, 3, 5, 7, ... is an ordered list of terms. The series 1 + 3 + 5 + 7 + ... is the sum of those terms — a completely different object. The sequence is a function mapping integers to values; the series is a number (or a divergent accumulation). Sigma notation makes the distinction explicit: the sequence is aₙ = 2n-1, while the series is Σaₙ. Always ask: are we listing, or are we adding?
Question 4 True / False
The expression Σᵢ₌₀⁵ i² has 5 terms because the index runs from 0 to 5.
TTrue
FFalse
Answer: False
The index runs through the values 0, 1, 2, 3, 4, 5 — that is 6 terms, not 5. When the index runs from a to b inclusive, the number of terms is b - a + 1. Σᵢ₌₀⁵ has 5 - 0 + 1 = 6 terms. This off-by-one error is extremely common with sigma notation and propagates into incorrect partial sum calculations.
Question 5 Short Answer
What is the difference between a sequence and a series, and why does that distinction matter for understanding infinite sums?
Think about your answer, then reveal below.
Model answer: A sequence is an ordered list of numbers (e.g., 1, 1/2, 1/4, ...); a series is the sum of the terms of a sequence (e.g., 1 + 1/2 + 1/4 + ...). The distinction matters because adding infinitely many terms does not necessarily produce infinity — it depends on whether the partial sums converge. The partial sums of a series form their own sequence, and asking whether that sequence converges to a finite limit is the central question of series theory.
A sequence can decrease toward zero without the series converging — the harmonic series 1 + 1/2 + 1/3 + ... diverges even though the terms go to 0. A series converges when the partial sum sequence has a finite limit. Understanding that a series is a sequence of partial sums — not just an infinite list — is what makes calculus-level convergence analysis possible.