<s>[INST] Prove by induction $3+3 \cdot 5+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$ My question is:
Prove by induction that $$3+3 \cdot 5+ 3 \cdot 5^2+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$$ whenever $n$ is a nonnegative integer. 
I'm stuck at the basis step. 
If I started with $1$. I get the right hand side is $18$ which is clearly not even close. It says prove shouldn't it be always true?
 [/INST] Proof by induction on $n$;
Set your base induction for $n = 1$; which gives you $18 = 18.$
Suppose your equation is true for $n$. We are interested to prove its correctness for $n+1.$ So
$3 + 3 \times 5 + \cdots + 3 \times 5^n + 3 \times 5^{n+1} = \frac{3 \times 5^{n+1} - 3}{4} + 3 \times 5^{n+1} = \frac{3 \times 5^{n+1} - 3 + 12 \times 5^{n+1} }{4} = \frac{3 ( 5^{n+1} + 4 \times 5^{n+1})  - 3}{4} = \frac{3 ( 5^{n+2} - 1)}{4}$
by using the induction steps. And we are done!
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