(1- 14)(1- 19)(1- 116)(1- 125)\cdots(1- 1225) = ?

Free Practice Question

PS · Problem Solving

655-705

Arithmetic

$(1-\frac{1}{4})(1-\frac{1}{9})(1-\frac{1}{16})(1-\frac{1}{25})\cdots(1-\frac{1}{225}) = ?$

Correct answer marked below

\frac{1}{75}

\frac{8}{75}

\frac{8}{15}

Correct

\frac{8}{225}

\frac{1}{5}

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Explanation

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Comments

Harsh1d ago

How do you know to use difference of squares here? I see the pattern but would've never thought to do that.

Bojust now

Look at the terms: 1 - 1/n^2 = (n-1)(n+1)/n^2. Then it telescopes.

TrintjeAug 19

I got 8/75 but that's not even an option? Wait, is it? Let me recalculate... I might have messed up the cancellation.

AbdulazizAug 26

Check the last term: n=15, so 1 - 1/225 = 224/225 = (14*16)/(15^2). The product telescopes to (1*16)/(2*15) = 16/30 = 8/15. So 8/75 is not it.

DevallaAug 23

This was a bit tricky but I loved the telescoping! Once you rewrite each term, it's so satisfying.

AntheaJul 3

I'm stuck on the last term. Is it supposed to be (1 - 1/225)? The pattern goes up to 15^2, right?

DaryaJul 6

Yes, 15^2=225, so the last term is 1 - 1/225. Then write it as (15-1)(15+1)/15^2.

What this question tests

Your fluency with the order of operations, fractions, decimals, and basic number sense — the foundation every quant question leans on.

Applying the order of operations out of sequence or rounding intermediate values before the final step.