Question

Find the sum of all natural numbers between 200 and 400 which are divisible by 7.

Answer 1

Answer:

The first term after 200 divisible by 7 is 203. The last term before 400 divisible by 7 is 399. The number of terms = ?

=> l = a + ( n - 1 ) d

=> 399 = 203 + ( n - 1 ) 7

=> 399 - 203 = ( n - 1 ) 7

=. 196 = ( n - 1 ) 7

=> 196 / 7 = ( n - 1 )

=> 28 = ( n - 1 )

=> n = 28 + 1 = 29

Hence the number of terms is 29.

Applying Sum formula we get,

$\implies S_n = \dfrac{n}{2} \: [ a + l ]\\\\\implies S_{29} = \dfrac{29}{2} \: [ 203 + 399 ]\\\\\implies S_{29} = \dfrac{29}{2} \times 602\\\\\implies S_{29} = 29 \times 301 \\\\\implies S_{29} = 8729$

Hence the sum of all the natural numbers divisible by 7 lying between 200 and 400 is 8729.

Answer 2

$\bf\large\boxed{\boxed{S_{29} = 8729}}$
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We need to find least number and greatest number divisible by 7 between 200 and 400.

Then we'll form an A.P with,

$a = least \: term \: divisible \: by \: 7$
$a_n \: or \: l$= $greatest \: term \: divisible \: by \: 7$
$d = 7$
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Now, divide 400 by 7.

If you don't get any remainder then 400 is divisible by 7 and hence largest divisible number but if you get a remainder (which you will), then subtract the remainder from 400 to get largest number.

In this case, we get remainder = 1
Subtracting remainder from 400 = 400-1 = 399

Hence, largest number divisible by 7 is 399

=> $\boxed{\mathsf{l = 399}}$ ✔️

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Now we'll divide 200 by 7.

This time if we get a remainder not equal to 0 then we'll subtract it from 200 and add 7.

The remainder we get is 4.

Now,

=> 200 - 4 + 7
=> 200 + 3
=> $\boxed{\mathsf{ a = 203}}$✔️
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Now,
We know that,

a = 203
l = 399
d = 7
n = ?
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Using the formula,

✅$\boxed{\mathbf{a_n = a + (n-1)d}}$✅

=> $\mathsf{399 = 203 + (n-1)7}$

=> $\mathsf{399 - 203 = (n-1)7}$

=> $\mathsf{196 = (n-1)7}$

=> $\mathsf{\frac{196}{7} = n - 1}$

=> $\mathsf{ n - 1 = 28}$

=> $\boxed{\mathsf{n = 29}}$✔️

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Now,
We need to calculate sum of all terms.

Using the formula,

✅$\boxed{\mathbf{S_n = \frac{n}{2} (a + l)}}$✅

=> $\mathsf{S_{29} = \frac{29}{2} (203 + 399)}$

=> $\mathsf{S_{29} = \frac{29}{2} \times 602}$

=> $\mathsf{S_{29} = 29 \times 301}$

=> $\boxed{\mathsf{S_{29} = 8729}}$✔️✔️

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Hence, Sum of All Natural Numbers divisible by 7 between 200 and 400 is 8729.
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