Question

Prove the following using induction:

(a) For all natural numbers n>2, 2n>2n+1

(b) For all positive integersn, 1^3+3^3+5^3+···+(2^n−1)^3=n^2(2n^2−1)

(c) For all positive natural numbers n,5/4·8^n+3^(3n−1) is divisible by 19

Answer 1

a)

i think you have written something wrong

you wrote

2n> 2n+1

which is clearly wrong, post the correct question again

b)

we check for n= 1

LHS = 1

RHS = 1^2 * (2*1^2 -1) = 1

suppose it is true for n = k

that is 1^3 + 3^3 + .. (2k -1) ^3 = k^2(2k^2−1)

we will show if it is true for n = k, then it is true for n = k+1

now

1^3 + 2^3 + .. (2^k -1)^3 + ((2(k+1) -1))^3 = k^2(2k^2−1) + ((2(k+1) -1))^3

= 2k^4 -k^2 + (2k+1)^3

= 2k^4 -k^2 + 8k^3 + 12 k^2 + 6k+1

= 2k^4 + 8k^3 +11 k^2 + 6k+1

= 2(k^4 + 4k^3 + 6 k^2 + 4k+1) - (k^2 +2k+1)

= 2 (k+1)^4 -(k+1)^2

= (k+1)^2 (2(k+1)^2 -1)

hence it is true for n = k+1

hence proved

similarly do c) part

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