+845 
for A i subbed in the values and expanded the brackets which gave me
15^n+1 - 8^n-1 - (120^n -8^n-1)
205^n then i divided that by 7 which gave me 7*15^n 7=k
B is what im confused about
so try for n =2
15^2 - 8^2-2 = 224 divisible 7 =32
assume true for n =k
15^k - 8^k-2
try for n=k+1
15^k+1 - 8^k-1
then i did 15*15^k - 8^k/8 (im not sure whether i can do this)
225^k - 1^k =224^k
224^k / 7 = 15^k
= 15^k * 7
but i dont think this is what im supposed to do please help thanks
+128399 b) Show that this is true for n = 2
15^2 - 8^(2 - 2) = 225 - 1 = 224 = 32(7)
Assume it is true for n = k.....that is
15^k - 8^(k - 2) is a multiple of 7
Prove it is true for k + 1
15^(k + 1) - 8^(k + 1 - 2)
15^(k + 1) - 8^( k - 1) note YEEEEEET that we can write this as
(14 + 1)^(k + 1) - (7 + 1)^(k - 1) use the binomial theorem and expand
[14^(k + 1) + C(k + 1, 1)*14^k + ... + 14 + 14^0* 1^(k + 1) ] -
[ 7^(k - 1 ) + C(k - 1, 1) * 7^(k - 2) + ....+7 + 7^0*1^ (k - 1) ]
The terms 14^0*1^(k +1) and 7^0*1^(k - 1) just equal 1 and will "cancel" with the subtraction of the second expansion from the first
So...we have this simplification....
[ 14^(k + 1) + C(k + 1, 1)* 14^k +...+ 14 - 7^(k - 1) - C(k - 1, 1)*7^(k - 2 ) -...- 7 ] =
[ (7 *2)^(k + 1) + C( k + 1,1)* (7*2)^k + ....+ 14 - 7^( k - 1) - C(k - 1, 1)*7^(k - 2) -....- 7 ]
Every term in the expansion simplification will be divisible by 7, hence the result is a multiple of 7
