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-Suppose you wish to prove the statement that follows using mathematical induction. $$4 + 9 + 14 + \ldots + ( 5 n - 1 ) = \frac { n } { 2 } ( 5 n + 3 ) , \text { for all positive integers } n$$ Let $S _ { n }$ be the statement $4 + 9 + 14 + \ldots + ( 5 n - 1 ) = \frac { n } { 2 } ( 5 n + 3 )$. Show that $S _ { 1 }$ is true.
A) Since $4 + 9 = 13 = \frac { 2 } { 2 } ( 5 ( 2 ) + 3 ) , S _ { 1 }$ is true.
B) Since $( 5 ( 1 ) - 1 ) = 4 , S _ { 1 }$ is true.
C) Since $4 + ( 5 ( 1 ) - 1 ) = \frac { 2 } { 2 } ( 4 + 5 ( 1 ) - 1 ) , S _ { 1 }$ is true.
D) Since $\frac { 1 } { 2 } ( 5 ( 1 ) + 3 ) = \frac { 1 } { 2 } ( 8 ) = 4 , S _ { 1 }$ is true.

Question

Quizplus · Accepted Answer

To verify $S_1$ is true, we substitute $n = 1$ into the given formula $\frac{n}{2}(5n + 3)$ and check if it equals the first term of the sequence, which is $4$. Substituting $n = 1$ gives $\frac{1}{2}(5(1) + 3) = \frac{1}{2}(8) = 4$, which matches the first term of the sequence, thus proving $S_1$ is true.

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