Question

Sum of all odd numbers between 2 and 190

Answer 1

Answer:

190-2=188

188÷2=94

(94)²=8836

Step-by-step explanation:

Answer 2

The sum of all odd numbers between 2 and 190 is 9024

Given :

All odd numbers between 2 and 190

To find :

The sum of all odd numbers between 2 and 190

Formula :

Sum of first n terms of an arithmetic progression

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

Where First term = a

Common Difference = d

Solution :

Step 1 of 3 :

Write down all odd numbers between 2 and 190

All odd numbers between 2 and 190 are 3 , 5 , 7 , . . . . , 189

This is an arithmetic progression

Step 2 of 3 :

Calculate number of terms

First term = a = 3

Common Difference = d = 5 - 3 = 2

Let there are n terms in the arithmetic progression

Then nth term of the AP = 189

a + ( n - 1 )d = 189

⇒ 3 + ( n - 1 ) × 2 = 189

⇒ 3 + 2n - 2 = 189

⇒ 2n + 1 = 189

⇒ 2n = 189 - 1

⇒ 2n = 188

⇒ n = 94

Step 3 of 3 :

Calculate sum of all odd numbers between 2 and 190

Number of terms = n = 94

∴ The sum of all odd numbers between 2 and 190

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

$\displaystyle \sf = \frac{94}{2} \bigg[(2 \times 3) + (94 - 1) \times 2 \bigg]$

$\displaystyle \sf = 47 \times \bigg[6 + 186 \bigg]$

$\displaystyle \sf = 47 \times 192$

$\displaystyle \sf = 9024$

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