Question

find the sum of odd numbers between 1 and 121​

Answer 1

SOLUTION :

TO DETERMINE

The sum of odd numbers between 1 and 121

FORMULA TO BE IMPLEMENTED

If in an arithmetic progression

First term = a

Common Difference = d

1. The n th term of the progression is

$= \sf{a + (n - 1)d \: }$

2. The sum of first n terms of the progression is

$= \displaystyle \sf{ \frac{n}{2} \bigg [ \: \: 2a + (n - 1)d \bigg] \: }$

EVALUATION

The odd numbers between 1 and 121 are

3, 5, 7, 9 , ......... , 119

This is an Arithmetic progression

Here the first term = a = 3

Common Difference = d = 5 - 3 = 2

Let 119 be the nth term of the AP

So

$\sf{a + (n - 1)d = 119 \: }$

$\implies \: \sf{3 + (n - 1)2 = 119 \: }$

$\implies \: \sf{ (n - 1)2 = 116 \: }$

$\implies \: \sf{ (n - 1) = 58\: }$

$\implies \: \sf{n = 59 \: }$

So there are 59 odd numbers between 1 and 121

Hence the sum of odd numbers between 1 and 121 is

$= \displaystyle \sf{ \frac{n}{2} \bigg [ \: \: 2a + (n - 1)d \bigg] \: }$

$= \displaystyle \sf{ \frac{59}{2} \bigg [ \: \: (2 \times 3) + (58 \times 2) \bigg] \: }$

$= \displaystyle \sf{ \frac{59}{2} \times 122 \: }$

$= \displaystyle \sf{ 59 \times 61 \: }$

$= 3599$

RESULT

The sum of odd numbers between 1 and 121

= 3599

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LEARN MORE FROM BRAINLY

the sum of the third and seventh term of an AP is 40 and the sum sixth and 14th terms is 70

Find the sum of first ten terms

https://brainly.in/question/22811954

Answer 2

The odd numbers between 1 and 121​ are:

3, 5, 7, ..........., 119

Here, first term (a) = 3, common difference (d) = 2 and Last term ($a_{n}$) = 119

Let nth be the term of an A.P. .

We have to find,  the sum of odd numbers between 1 and 121​.

Solution:

We know that:

The nth term of an A.P. ($a_{n}$).

$a_{n}$ = a + (n -1)d

∴ a + (n -1)d = 119

⇒ 3 + (n - 1)2 = 119

⇒ (n - 1)2 = 119 - 3 = 116

⇒ n - 1 = $\dfrac{116}{2}$

⇒ n - 1 = 58

⇒ n = 58 + 1

⇒ n = 59

We also know that:

The sum of nth term of an A.P.,

$S_{n} =\dfrac{n}{2} [a+a_n]$

∴ The sum of 59th term of an A.P.,

$S_{59} =\dfrac{59}{2} [3+119]$

⇒ $S_{59} =\dfrac{59}{2} [122]$

⇒ $S_{59} =59\times 61$

⇒ $S_{59} =3599$

∴ The sum of odd numbers between 1 and 121 = 3599

Thus, the sum of odd numbers between 1 and 121 is "equal to 3599".