Use mathematical induction to solve for all positive integers n. $5 + 9 + 13 + 17 + \ldots + ( 4 n + 1 ) = ?$
A) $n ( 9 n + 3 )$
B) $\frac { n ( 2 n + 3 ) } { 2 }$
C) $n ( n + 3 )$
D) $\frac { n ( n + 3 ) } { 9 }$
E) $n ( 2 n + 3 )$

Question

Use mathematical induction to solve for all positive integers n.   $5 + 9 + 13 + 17 + \ldots + ( 4 n + 1 ) = ?$  
A) $n ( 9 n + 3 )$
B) $\frac { n ( 2 n + 3 ) } { 2 }$
C) $n ( n + 3 )$
D) $\frac { n ( n + 3 ) } { 9 }$
E) $n ( 2 n + 3 )$

Quizplus · Accepted Answer

The sequence is an arithmetic sequence with the first term 5 and common difference 4. The sum of the first n terms of an arithmetic sequence is given by $S_n = \frac{n}{2} (a + l)$, where $a$ is the first term and $l$ is the last term. Here, $l = 4n + 1$. Substituting these values, we get $S_n = \frac{n}{2} (5 + (4n + 1)) = n(2n + 3)$.

Use Mathematical Induction to Solve for All Positive Integers N 5+9+13+17+…+(4n+1)=?5 + 9 + 13 + 17 + \ldots + ( 4 n + 1 ) = ?5+9+13+17+…+(4n+1)=?

Use Mathematical Induction to Solve for All Positive Integers N $5 + 9 + 13 + 17 + \ldots + ( 4 n + 1 ) = ?$