Question

Find the sum of all the natural numbers between 10 and 200 which are divisible by 7.
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Answer 1

Question :-

• Find the sum of all the natural numbers between 10 and 200 which are divisible by 7.

Answer

Given :-

• Natural Numbers between 10 and 200.

To find :-

• Sum of all numbers divisible by 7 between 10 to 200.

Formula used :

nth term of an AP is

$\bf \:a_n = a + (n - 1) \times d$

Sum of n terms of an AP is

$\bf \:S_n = \dfrac{n}{2} (2a + (n - 1)d)$

where,

• a = first term of an AP
• d = Common Difference of an AP

Solution :-

Let we first find the series of number from 11 to 199 which are divisible by 7.

14 + 21 + 28 + ......... + 196

Here, First term, a = 14

Common Difference, d = 7

nth term = 196

Using the formula of nth term

$\bf \:a_n = a + (n - 1) \times d$

On substituting the values, we get

⇛ 196 = 14 + (n - 1) × 7

⇛ 196 - 14 = (n - 1) × 7

⇛ 182 = (n - 1) × 7

⇛ 26 = n - 1

⇛ n = 27.

Now, Sum of 27 terms of an AP series is

$\bf \:S_n = \dfrac{n}{2} (2a + (n - 1)d)$

On substituting the values, we get

$\bf \:S_{27} = \dfrac{27}{2} (2 \times 14 + (27 \: - 1) \times 7)$

$\bf\implies \:S_{27} = \dfrac{27}{2} (28 + 26 \times 7)$

$\bf\implies \:S_{27} = \dfrac{27}{2} (28 + 182)$

$\bf\implies \:S_{27} = \dfrac{27}{2} \times 210$

$\bf\implies \:S_{27} = 27 \times 105$

$\bf\implies \:S_{27} = 2835$

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Answer 2

Step-by-step explanation:

a=14

d=7

L=196

=a+ (n-1) d= 196

=14 + (n-1) (7) = 196

= n-1) (7) = 182

= n - 1 = 26

= n= 27

An = n/2 ( a + L)

= 27/2 ( 14 + 196 )

= 27 multiply 210/ 2

= 27 multiply 105

= 2835