<s>[INST] Prove by induction that $\forall n\geq 1,\ 7\mid 3^{2n+1} + 2^{n-1}$ Prove by induction that $$7 \mid 3^{2n+1} + 2^{n-1},\ \forall n\geq 1$$ 
Base case $n=1$:
$$3^{2 × 1+1} + 2^{1-1} = 28.$$
Induction:
$$P(k): 3^{2k+1} + 2^{k-1},\ P(k+1): 3^{2(k+1)+1} + 2^{(k+1)-1}.$$
$$3^{2k+3} + 2^k = 9 \times 3^{2k+1} + 2^{k-1} \times 2 = 7 \times 3^{2k+1} + 2 \times 3^{2k+1} + 2^{k-1} \times 2.$$
Where do I go from here?
 [/INST] $$P(k)=3^{2k+1}+2^{k-1}=7 \lambda (let)$$
Now, For $(k+1)$,
$$9*3^{2k+1}+2^{k-1}.2$$
Put $2^{k-1}$ form first equation i wrote
$$7.(3^{2k+1}+2\lambda)$$
Hope your doubt is solved?
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