Sum to a Billion

Algebra Level 4

1 0 9 + ∑ k = 0 1 0 9 k ( − 1 ) k = ? \large 10^9 + \sum_{k=0}^{10^9} k (-1)^k = \, ?


The answer is 1500000000.

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Kevin Lee by Kevin Lee , 31, USA

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1 solution

Chew-Seong Cheong
Jul 19, 2016

Relevant wiki: Algebraic Manipulation - Rearranging

1 0 9 + ∑ k = 0 1 0 9 ( − 1 ) k k = 1 0 9 + 0 − 1 + 2 − 3 + 4 − 5 + 6 − . . . − ( 1 0 9 − 1 ) + 1 0 9 = 1 0 9 − 1 + 2 − 3 + 4 − 5 + 6 − . . . − ( 1 0 9 − 1 ) + 1 0 9 = 1 0 9 + 1 + 1 + 1 + . . . 1 ⏟ 5 × 1 0 8 × 1 = 1 0 9 + 5 × 1 0 8 = 15 × 1 0 8 = 1500000000 \begin{aligned} 10^9 + \sum_{k=0}^{10^9} (-1)^k k & = 10^9 + 0 - 1 + 2 - 3 + 4 - 5 + 6 - ... - (10^9 - 1) + 10^9 \\ & = 10^9 \color{#3D99F6}{- 1 + 2} \color{#D61F06}{- 3 + 4} \color{#3D99F6}{- 5 + 6} - ... \color{#D61F06}{- (10^9 - 1) + 10^9} \\ & = 10^9 + \underbrace{\color{#3D99F6}{1} + \color{#D61F06}{1} + \color{#3D99F6}{1} + ... \color{#D61F06}{1}}_{5 \times 10^8 \times 1} \\ & = 10^9 + 5 \times 10^8 \\ & = 15 \times 10^8 = \boxed{1500000000} \end{aligned}

19 Helpful 0 Interesting 0 Brilliant 0 Confused

Clean-cut telescoping sequence. Nice!

Ian Limarta - 4 years, 11 months ago

the system needs to be changed. i wrote 15E08 it did n't accept my answer i thought my answer was wrong but i was right.

A Former Brilliant Member - 4 years, 10 months ago

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I did that also. I remember it used to work. I think the problem setter have to set it. Problem should also change the answer option so as to avoid careless mistake.

Chew-Seong Cheong - 4 years, 10 months ago

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