Select the right expression for the following sum and prove that your statement is true for all natural numbers $n$.
$P _ { \mathrm { n } } : 3 + 6 + 9 \ldots 3 n$
A) $\quad \frac { n ( 3 n + 1 ) } { 2 }$
B) $\quad \frac { 3 n ( n + 1 ) } { 4 }$
C) $\quad \frac { 3 n ( n + 1 ) } { 2 }$
D) $\frac { n ( 3 n + 1 ) } { 4 }$
E) none of the above

Question

Quizplus · Accepted Answer

The given sum is an arithmetic progression with first term $3$ and common difference $3$. Therefore, the sum of the first $n$ terms is equal to $\frac{n}{2}(2\cdot 3+(n-1)3)=\frac{3n^2}{2}+\frac{3n}{2}$. Factoring $\frac{3}{2}$, we get $\frac{3n(n+1)}{2}$, which matches answer choice C. To prove that this statement is true for all natural numbers, we used the formula for the sum of an arithmetic series, which holds for all positive integers $n$. Thus, the answer is $\boxed{	ext{C}}$.

Select the Right Expression for the Following Sum and Prove nnn

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