Question

The sum of 25 terms of an A.P., whose all the terms are
natural numbers, lies between 1900 and 2000 and its 9th
term is 55. Then the first term of the A.P. is​

Answer 1

Answer:

Given:

• The sum of 25 terms of an A.P., whose all the terms are natural numbers, lies between 1900 and 2000
• 9th term is 55 => a+8d=55

step by step explanation:

let first term of an AP be a common difference be d

we have a+8d=55........(1)

Also 225(a+a+24d)=sum of 25 terms of an AP

1900<225(2a+24d)<2000

1900<25(a+12d)<2000

76<a+12d<80

76<55−8d+12d<80

21<4d<25⇒421<d<425

put d=6  in (1) we have

 

a+48=55⇒a=7.

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Answer 2

$\huge\bold{\underline{\sf{\pink{AnSwer :-}}}}$

Let the first term of an AP be a common difference $d.$

It is given that,$a + 8d = 55.....(1)$

Also,

$\frac{25}{2} (a + a + 24d) = sum \: of \: 25 \: terms \: of \: an \: ap$

$\implies1900< \frac{25}{2}(2a + 24d)< \:2000$

$\implies 1900< \: 25 \: (a+12d)< \: 2000$

$\implies 76< \: a+12d< \: 80$

$\implies76< \: 55 - 8d+12d< \: 80$

$\implies21< \: 4d< \: 25 \\\implies \: \frac{21}{4} < \: d \: < \frac{25}{4}$

Substitute $d = 6$in equation (1)

Now, we have,

$\implies \: a+48=5 \\ \implies \: a=7$