<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] As you want to prove for $n$ nonnegative and integer, your basis is $n=0$
You just have to prove that $\frac{3(5^{0+1} -1)}{4} = 3$ and it is true, because 
$\frac{3(5^{0+1} -1)}{4} = \frac{3(5 -1)}{4} = \frac{3.4}{4} = 3$
If you want to dofor $n = 1$:
For $n=1$, you have that: 
$3 + 3.5 = 18$ and $\frac{3(5^{1+1} -1)}{4} = \frac{3(5^2 -1)}{4} = 18$
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