Chapter 11: Q10. (page 586)

Find the sum of each arithmetic series.
$\sum_{n = 1}^{7} (2 n + 1)$

Short Answer

Expert verified

The sum of the arithmetic series is63.

Step by step solution

Step 1. Given Information.

Given arithmetic series is $\sum_{n = 1}^{7} (2 n + 1)$ .

Step 2. Calculation.

From the summation, the number of terms is $n = 7$

The first term is given by plugging $n = 1$ :

$\begin{array}{l} a_{n} = 2 n + 1 \\ a_{1} = 2 (1) + 1 \\ a_{1} = 2 + 1 \\ a_{1} = 3 \end{array}$

The last term is given by plugging $n = 7$ :

$\begin{array}{l} a_{n} = 2 n + 1 \\ a_{7} = 2 (7) + 1 \\ a_{7} = 14 + 1 \\ a_{7} = 15 \end{array}$

The sum $S_{n}$ of the first n terms of an arithmetic series is given by

$S_{n} = \frac{n}{2} (a_{1} + a_{n})$

Here, $a_{1} = 3, n = 7, a_{n} = 15$

Plugging the values:

$\begin{array}{l} S_{n} = \frac{n}{2} (a_{1} + a_{n}) \\ S_{7} = \frac{7}{2} (3 + 15) \\ S_{7} = \frac{7}{2} (18) \\ S_{7} = 7 (9) \\ S_{7} = 63 \end{array}$

Step 3. Conclusion.

Hence, the sum is 63.

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Find the sum of each arithmetic series.
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Step by step solution

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Step 2. Calculation.

Step 3. Conclusion.

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Find the sum of each arithmetic series.
$\sum_{n = 1}^{7} (2 n + 1)$