Prove by induction $4$ is a divisor of $(3^n +2n-1)\; ,n\ge1$

Question

Prove by induction $4$ is a divisor of $(3^n +2n-1)\; ,n\ge1$

My idea:

for $n=1$ ,$3^n+2n-1=4$

therefor true for $n=1$

now suppose this is true for $n=k$

i.e,. $4$ is the divisor of $3^k+2k-1$

we have to prove for $n=k+1$

so consider $ 3^{k+1}+2(k+1)-1=3.3^k+2k+1$ how to processed from here

Answer 1

Hint: $3*3^k + 2k+1 = 3(3^k + 2k-1)-4k+4$

Answer 2

To do this without magic, look at the difference of two consecutive terms.

This is

$(3^{n+1} +2(n+1)-1)-(3^n +2n-1) =3^{n+1}-3^n+2 =3^n(3-1)+2 =2(3^n+1) $.

Since $3^n$ is odd, $3^n+1$ is even so $2(3^n+1)$ is divisible by $4$.

This establishes the induction step.