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Chapter 12: Q. 37 (page 816)

Find each sum.
$1 + 3 + 5 + . . . . . . . . . + (2 n - 1)$

Short Answer

Expert verified

The solution is $S_{n} = n^{2}$ .

Step by step solution

01

Step 1. Given information

$1 + 3 + 5 + . . . . . . . . . + (2 n - 1)$

02

Step 2. Calculation

We have, $1 + 3 + 5 + . . . . . . . . . + (2 n - 1)$

So,localid="1649220172843" $a_{1} = 1, d = 2, a_{n} = (2 n - 1)$

Now use formula -

localid="1649220724840" $S_{n} = \frac{n}{2} (a_{1} + a_{n})$

localid="1649220732601" $= \frac{n}{2} [1 + (2 n - 1)]$

$= \frac{n}{2} (2 n)$

$= n^{2}$

Therefore, the sum islocalid="1649220855825" $S_{n} = n^{2}$ .

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Most popular questions from this chapter

In Problems 37–50, a sequence is defined recursively. Write down the first five terms.

$a_{1} = \sqrt{2}; a_{n} = \sqrt{2 + a_{n - 1}}$

In Problems 51–66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.

$8 + 4 + 2 + . . . .$

In Problems 37–50, a sequence is defined recursively. Write down the first five terms.

$a_{1} = A; a_{n} = a_{n - 1} + d$

In Problems 51–66, determine whether each infinite geometric series converges or diverges. If it converges, find its sum.

\sum_{k = 1}^{\infty} 5 {(\frac{1}{4})}^{k - 1}

In Problems 61–70, express each sum using summation notation.

$1 + 3 + 5 + 7 + . . . . + [2 (12) - 1]$

See all solutions

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