Evaluate the Summation sum from n=3 to 6 of 2n^2+3

$\sum_{n = 3}^{6} 2 n^{2} + 3$

Step 1

Expand the series for each value of $n$ .

$2 \cdot 3^{2} + 3 + 2 \cdot 4^{2} + 3 + 2 \cdot 5^{2} + 3 + 2 \cdot 6^{2} + 3$

Step 2

Simplify.

Raise $3$ to the power of $2$ .

$2 \cdot 9 + 3 + 2 \cdot 4^{2} + 3 + 2 \cdot 5^{2} + 3 + 2 \cdot 6^{2} + 3$

Multiply $2$ by $9$ .

$18 + 3 + 2 \cdot 4^{2} + 3 + 2 \cdot 5^{2} + 3 + 2 \cdot 6^{2} + 3$

Add $18$ and $3$ .

$21 + 2 \cdot 4^{2} + 3 + 2 \cdot 5^{2} + 3 + 2 \cdot 6^{2} + 3$

Raise $4$ to the power of $2$ .

$21 + 2 \cdot 16 + 3 + 2 \cdot 5^{2} + 3 + 2 \cdot 6^{2} + 3$

Multiply $2$ by $16$ .

$21 + 32 + 3 + 2 \cdot 5^{2} + 3 + 2 \cdot 6^{2} + 3$

Add $32$ and $3$ .

$21 + 35 + 2 \cdot 5^{2} + 3 + 2 \cdot 6^{2} + 3$

Add $21$ and $35$ .

$56 + 2 \cdot 5^{2} + 3 + 2 \cdot 6^{2} + 3$

Raise $5$ to the power of $2$ .

$56 + 2 \cdot 25 + 3 + 2 \cdot 6^{2} + 3$

Multiply $2$ by $25$ .

$56 + 50 + 3 + 2 \cdot 6^{2} + 3$

Add $50$ and $3$ .

$56 + 53 + 2 \cdot 6^{2} + 3$

Add $56$ and $53$ .

$109 + 2 \cdot 6^{2} + 3$

Raise $6$ to the power of $2$ .

$109 + 2 \cdot 36 + 3$

Multiply $2$ by $36$ .

$109 + 72 + 3$

Add $72$ and $3$ .

$109 + 75$

Add $109$ and $75$ .

$184$