Question 7

The terms would be 2^15, 2^16, and 2^17

Just apply the answer choices to the numbers we are already given.



A. 2^18 (4th number would be sum of first 3) does 8 = 1 + 2 + 4? No
B. 3(2^17) (3 times the 3rd number) Does 3(4) = 1 +2 + 4? No
C. 7(2^16) (7 times the 2nd number) Does 7(2) = 1 + 2 + 4? No
D. 3(2^16) (3 time 2nd number) Does 3(2) = 1 +2 +4? No
E. 7(2^15) (7 times first number) Does 7(1) = 1 +2 +4? Yes

To check: does 7(4) = 4 + 8 + 16? Yes

So answer is E.

Just my approach, but I am sure there is a better one.

My approach -

2^15 + 2^16 + 2^17 = 2^15 (1 + 2 + 2^2)

= 2^15 (7) = E

The first step is to realize that the terms are 2^15, 2^16, and 2^17

next I applied a technique I learned (quite recently) from the incredibly knowledgeble Cramya.

2 is a special number in that 2^x + 2^x = 2^x + 1

therefore

2^15 + 2^16 + 2^17 =

2^15 + 2^15 + 2^15 + 2^16 + 2^16 =

2^15 + 2^15 + 2^15 +2^15 + 2^15 + 2^15 + 2^15 =

7(2^15) = answer E

** thanks Cramya! keep the short cuts coming **

I am sure ! cramya ! gave this nice explanation sometime back on the repeating 2 powers sequence and playing around with them

so following the sequence 2^15, 2^16, and 2^17 will be 16th,17th and 18th

adding up we get

2^15(1+2+4)

IMO=7*2^15

Vishu

This is a geometric series.
The n-th term of a geometric sequence with initial value a1 and common ratio r is given by

an = a1r^n-1
a1 = 1
r = 2

a16 = 1 (2)^ 15
a17 = 1(2) ^16
a18 = 1(2) ^17

a16 + a17 + a18 =
2^15 + 2^16 + 2^17
Factor out 2^15
2^15(1 + 2 + 2^2)
2^15(7)
My answer would be E.