a+a^2+a^3+a^4+a^5 = a(1-a^5)/(1-a) This formula is the sum of five terms of a geometric sequence with first term a and reason a, i.e each term is obtained from the previous one multiplying by a. Now just substitute a=2^3 and we get 2^3(1-2^15)/(1-2^3) = -8/7(1-2^15) (1) We now just have to calculate 2^15 = 2^6*2^6*2^3 = 64*64*8 = 4096*8 = 32768 Substitute 2^15=32768 in (1) and you get -8/7*(1- 32768) =(-8)*(-4681) =37448
But still you do not need a calculator to directly calculate the powers of 2 and sum to get the result. 2^3=8 2^6=2^3*2^3=8*8=64 2^9=2^6*2^3=64*8=512 2^12=2^6*2^6=64*64=4096 2^15=2^12*2^3=4096*8=32768 So the sum is 8+64+512+4096+32768 = 37448
a+a^2+a^3+a^4+a^5 = a(1-a^5)/(1-a)
This formula is the sum of five terms of a geometric sequence with first term a and reason a, i.e each term is obtained from the previous one multiplying by a.
Now just substitute a=2^3 and we get
2^3(1-2^15)/(1-2^3) =
-8/7(1-2^15) (1)
We now just have to calculate
2^15 = 2^6*2^6*2^3 = 64*64*8 = 4096*8
= 32768
Substitute 2^15=32768 in (1) and you get
-8/7*(1- 32768)
=(-8)*(-4681)
=37448
Well, I must admit your method is better though, because you only have to calculate 9*64*65+8=37448
But still you do not need a calculator to directly calculate the powers of 2 and sum to get the result.
2^3=8
2^6=2^3*2^3=8*8=64
2^9=2^6*2^3=64*8=512
2^12=2^6*2^6=64*64=4096
2^15=2^12*2^3=4096*8=32768
So the sum is
8+64+512+4096+32768 =
37448
Whatever method we choose no calculator is needed but your method requires the least calculations, so well done sir!
hi