Question

The sum of the first three
terms of an Arithmetic
Progression (A.P.) is 45 and the
product of the first and third
term is 21. Find the first term
and the common difference.​

Answer 1

Answer:

answer for the given problem is given

Answer 2

$\huge\bold{\gray{\sf{Answer:}}}$

Explanation:

Given :

The sum of the first three

terms of an Arithmetic

Progression (A.P.) is 45 and the

product of the first and third

term is 21.

To Find :

Find the first term

and the common difference.

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ㅤㅤㅤㅤㅤㅤㅤㅤㅤㅤ

Let the first term be 'a-d'

Second term be 'a'

Third term be 'a+d'

Given that sum of first three term is 45

Then ,a-d+a+a+d=45

$= > 3a = 45$

$\bold{\boxed{\red{a = 15}}}$

Given that product of first and third term is 21

$= > (a - d)(a + d) = 21$

Here, identity is used:-

$\bold{\boxed{\purple{(a + b)(a - b) = {a}^{2} - {b}^{2} }}}$

$= > {a}^{2} - {d}^{2} = 21$

$= > {(15)}^{2} - {d}^{2} = 21$

$= > 225 - {d}^{2} = 21$

$= > - {d}^{2} = 21 - 225 = - 204$

$= > {d}^{2} = 204$

$= > d = \sqrt{204}$

$= > d = \sqrt{2 \times 2 \times 17 \times 3}$

$\bold{\boxed{\red {d = 2 \sqrt{51} }}}$

First term of an a.p=>a-d=15-2√51

Second term of an a.p is =>15

Third term of an a.p is =>a+d=15+2√51

$\bold{Hence \:,the \:A.P \:are :-}$

$\bold{\boxed{15 - 2 \sqrt{51} ,15,15 + 2 \sqrt{51} ...∞}}$