2015 Countdown Problem #11: Find the Largest - Part II

Calculus Level 3

Which is the largest?

A. 2015 32 + 2015 3 2 2 + 2015 3 2 3 + . . . + 2015 3 2 2014 + 2015 3 2 2015 \frac{2015}{32}+\frac{2015}{32^2}+\frac{2015}{32^3}+...+\frac{2015}{32^{2014}}+\frac{2015}{32^{2015}}

B. 0.6 − 0.06 + 1.2 − 0.12 + 1.8 − 0.18 + . . . + 8.4 − 0.84 + 9.0 − 0.90 0.6-0.06+1.2-0.12+1.8-0.18+...+8.4-0.84+9.0-0.90

C. 45 + 15 + 5 + 5 3 + 5 3 2 + . . . + 5 3 14 + 5 3 15 45+15+5+\frac{5}{3}+\frac{5}{3^2}+...+\frac{5}{3^{14}}+\frac{5}{3^{15}}

D. 1 + 2 + 3 + . . . + 10 + 11 1+2+3+...+10+11

This problem is part of the set 2015 Countdown Problems .

D B A C

Wee Xian Bin by Wee Xian Bin , 25, Singapore

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1 solution

Wee Xian Bin
Dec 13, 2014

For (a):

Sum = 2015 32 + 2015 3 2 2 + 2015 3 2 3 + ⋯ + 2015 3 2 2014 + 2015 3 2 2015 < 2015 ( 3 2 − 1 3 2 − 1 − 1 ) = 2015 31 = 65 \frac{2015}{32}+\frac{2015}{32^2}+\frac{2015}{32^3} +⋯+\frac{2015}{32^{2014}} +\frac{2015}{32^{2015}} <2015(\frac{32^{-1}}{32^{-1}-1})=\frac{2015}{31}=65

Hence the sum (a) < 65 < 65

For (b):

Sum = 15 2 ( 2 × 0.6 + ( 15 − 1 ) × 0.6 ) + 15 2 ( 2 × − . 006 + ( 15 − 1 ) × − 0.06 ) \frac{15}{2} (2×0.6+(15-1)×0.6)+\frac{15}{2} (2×-.006+(15-1)×-0.06) = 72 − 7.2 = 64.8 < 65 = 72-7.2 = 64.8 < 65

For (c):

Sum > 45 + 15 + 5 + 5 3 > 66 >45+15+5+\frac{5}{3}>66

For (d):

Sum = 11 × 12 2 = 66 \frac{11×12}{2}=66

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