A senior high school competition-level trouble worrying amount and also series

Offered the amount of first $N$ components of series $A$: $S_{n} = n^{2} + 3n + 4$.
Calculate $A_{1} + A_{3} + \cdots + A_{21}$.

I recognize this trouble can be taken on by meticulously computing each value of the series. Yet I question what are the far better means to address it.

Many thanks beforehand!

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2019-05-07 00:27:03
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Answers: 2

\begin{equation*} A_n=S_n-S_{n-1}=2n+2. \end{equation*}

Now one needs to locate an amount of a math development $4+8+\dots+44$.

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2019-05-08 19:42:55
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Determining a wonderful kind for $A(n)$ is an excellent start. This is $S(n) - S(n-1) = 2n + 2 $. Currently the wanted amount is

\begin{equation*} \sum_{k= 1}^{10} A(2k-1) = \sum_{k=1}^{10} 4k = 4 \ast 10 \ast 11 / 2 = 220 .~_{\square} \end{equation*}

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2019-05-08 19:41:15
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