The sequence S is defined as $S_n = (n + 1)!$ for all integers $n ≥ 1$ . For example, $S_3 = 4! = (4)(3)(2)(1)$ . Which of the following is equivalent to the difference between $S_{100}$ and $S_{99}$ ?

Answer Choices

Correct answer marked below

101!

100!

99^2(98!)

100^2(99!)

Correct

(100!)^2

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Comments

ChristianJun 9

I initially thought it was 100! but then realized S100 is 101! and S99 is 100!, so factoring out 100! gives 100!(101-1) = 100! * 100. That's 100*100!, which equals 100^2 * 99!. Answer D.

NguyenJun 18

Wait, how did you get from 100*100! to 100^2 * 99!? I'm stuck there.

OjasJul 11

This one was straightforward once I wrote out the factorials. I just did S100 - S99 = 101! - 100! and factored. Took me a minute to see the trick with 99! but it's a nice little problem.

BrianJun 26

The wording 'difference between S100 and S99' could be ambiguous? Is it S100 - S99 or S99 - S100? Luckily the answer choices are all positive, so it must be the absolute difference. I got D.

TimJun 30

Yeah, I think they mean S100 - S99 since it's the difference between the larger and smaller. But good point.

What this question tests

What it tests

Your ability to spot and extend arithmetic and geometric sequences and work with recursive definitions.

Common trap

Off-by-one errors in indexing (a₀ vs a₁) or confusing an arithmetic difference with a geometric ratio.

The sequence S is defined as Sn=(n+1)!S_n = (n + 1)!Sn​=(n+1)! for all integers n≥1n ≥ 1n≥1. For example, S3=4!=(4)(3)(2)(1)S_3 = 4! = (4)(3)(2)(1)S3​=4!=(4)(3)(2)(1). Which of the following is equivalent to the difference between S100S_{100}S100​ and S99S_{99}S99​?