The sum $1^{2}-2 \cdot 3^{2}+3 \cdot 5^{2}-4 \cdot 7^{2}+5 \cdot 9^{2}-\ldots+15 \cdot 29^{2}$ is _________.
Answer (integer)
6952
Solution
$S=1^{2}-2.3^{2}+3.5^{2}-4.7^{2}+\ldots \ldots+15.29^{2}$
<br/><br/>Separating odd placed and even placed terms we get
<br/><br/>$$
\begin{aligned}
& \mathrm{S}=\left(1.1^2+3.5^2+\ldots .15 .(29)^2\right)-\left(2.3^2+4.7^2\right. \\
& +\ldots .+14 .(27)^2
\end{aligned}
$$
<br/><br/>$$
\begin{aligned}
& =\sum_{r=1}^{8}(2 r-1)(4 r-3)^{2}-\sum_{r=1}^{7} 2 r(4 r-1)^{2} \\\\
& =\sum_{r=1}^{8} (32 r^{3}-64 r^{2}+42 r-9)-2\sum_{r=1}^{7} 16 r^{3}-8 r^{2}+r \\\\
& =32 \times 36^{2}-64 \times 204+1512-72 \\\\
& -2\left(16 \times 28^{2}-1120+28\right) \\\\
& =6592
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Sequences and Series · Topic: Arithmetic Progression
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