Question

The sum of first three terms of an A.P is 45 and the sum of there squares is 693. If common difference is positive, Then what is its fourth term?

Answer 1

Let three terms are (a - d) , a , (a + d) , here a and d are first term and common difference of an AP.
A/C to question,
Sum of three terms = 45
(a - d) + a + (a + d) = 45
3a = 45 ⇒a = 15 ---------(1)

Again, sum of their square = 693
(a - d)² + a² + (a + d)² = 693
⇒a² + d² - 2ad + a² + a² + d² + 2ad = 693
⇒ 3a² + 2d² = 693
⇒ 3(15)² + 2d² = 693
⇒ 3 × 225 + 2d² = 693 [ from equation (1)
⇒675 + 2d² = 693
⇒ d² = 9 ⇒ d = ±3


∴numbers are [ for d = 3 ] : (a - d) = 15 - 3 = 12
a = 15
and (a + d ) = 15 + 3 = 18
Hence, numbers are 12, 15 , 18

[ Well you can take d = -3 , number will be same , just arrangement is opposite like 18, 15 , 12 ]

Answer 2

heya..!!!!

let the three terms be (a - d),a (a + d)

case(1)

a - d + a + a + d = 45

=> 3a = 45

=> a = 45/3

=> a = 15---(1)

case (2)

(a - d)² + a² + (a + d)² = 693

=> a ² + d² - 2ad + a² + a² + d² + 2ad = 693

=> 3a² + 2d² = 693

=> 3(15)² + 2d² = 693

=> 675 + 2d² = 693

=> 2d² = 18

=> d² = 18/2 = 9

=> d = +-3

(15 - 3) = 12
a = 15
(15 + 3) = 18

12 , 15 , 18

next term = (18 + d) = (18 + 3) = 21

hence, option (3) is correct