prove by induction 10^n+3.4^n+2+5 IS DIVISIBLE BY 9
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let n=1
so, 10+3*4³+5=207
207/9=23 hence divisible
let 10ⁿ+3.4ⁿ⁺²+5 is divisible by 9
then consider
10ⁿ⁺¹+3.4ⁿ⁺³+5
10.10ⁿ+3.4.4ⁿ⁺²+5=(10ⁿ+3.4ⁿ⁺²+5)+9*(10ⁿ+4ⁿ⁺²) individually both are divisible by 9
hence proved
so, 10+3*4³+5=207
207/9=23 hence divisible
let 10ⁿ+3.4ⁿ⁺²+5 is divisible by 9
then consider
10ⁿ⁺¹+3.4ⁿ⁺³+5
10.10ⁿ+3.4.4ⁿ⁺²+5=(10ⁿ+3.4ⁿ⁺²+5)+9*(10ⁿ+4ⁿ⁺²) individually both are divisible by 9
hence proved
kinkyMkye:
then assume its true for n
and prove its true for n+1 also
so the number for n+1 can be writein such a way that
(10^(n+1)+3.4^(n+1+2)+5 =
10*10^n+4*3*4^(n+2)+5
(10^n+9.10^n)+(3*4^(n+2)+3*3*4^(n+2))+5
10^n+3*4^(n+2)+5 + (9.10^n+9*4^(n+2))
we already assumed 10^n+3*4^(n+2)+5 id divisible by 9
(9.10^n+9*4^(n+2))=9 (10^n+4^(n+2)) hence divisible by 9
so the number is always divisible
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