Question

8,20,64,125,162___? Find missing number

Answer 1

Let f(n)=an⁴+bn³+cn²+dn+e

f(0)=e=8;

f(1)=a+b+c+d+e=20, so a+b+c+d=12;

f(2)=16a+8b+4c+2d=64-8=56;

f(3)=81a+27b+9c+3d=125-8=117;

f(4)=256a+64b+16c+4d=162-8=154;

A=f(2)-2f(1)=14a+6b+2c=32, 7a+3b+c=16;

B=f(3)-3f(1)=78a+24b+6c=81, 26a+8b+2c=27;

C=f(4)-4f(1)=252a+60b+12c=106, 126a+30b+6c=53;

B-2A=12a+2b=-5;

C-3B=48a+6b=-28;

(C-3B)-3(B-2A)=12a=-13, a=-13/12; -13+2b=-5, b=4; -91/12+12+c=16, c=16+91/12-12=(192+91-144)/12=139/12; d=12-(-13/12+4+139/12)=12-29/2=-5/2.

f(n)=-13n⁴/12+4n³+139n²/12-5n/2+8.

So n=5 produces f(5)=108, the next term in the series according to the formula (all other terms fit).

But this is only one of several solutions.