Question

The sum of 5 numbers in ap is 30 and the sum of their squares is 190. which of the following is the third term?

Answer 1

hii dear.....
good night

Answer 2

Solution :-

Let the 5 numbers in the given A. P. be (a - 2d), (a - d), (a), (a + d) and (a + 2d)

Sum of these is 30.

(a - 2d) + (a - d) + (a) + (a + d) + (a + 2d) = 30

5a = 30

a = 30/5

a = 6

As sum of their squares is 190.

Now, putting the value of a in the above.

(6 - 2d)² + (6 - d)² + (6)² + (6 + d)² + (6 + 2d)² = 190

⇒ (4d² -24d + 36) + (d² - 12d + 36) + (36) + (d² + 12d + 36) + (4d² + 24d + 36) = 190

⇒ 4d² + d² + d² + 4d² + 36 + 36 + 36 + 36 + 36 = 190

⇒ 10d² = 180 = 190

⇒ 10d² = 190 - 180

⇒ 10d² = 10

⇒ d² = 10/10

⇒ d² = 1

⇒ d =  √1

⇒ d = 1

So, the 5 numbers are -

1) (6 - 2d)

= 6 - 2*1

= 4

2) (6 - d) 

= 6 - 1

= 5

3) a = 6

4) (6 + d)

= 6 + 1

= 7

5) (6 + 2d)

= 6 + 2*1

= 6 + 2

= 8

Hence, the five numbers are 4, 5, 6, 7 and 8

3rd = a + (n - 1)d

= 6 + (3 - 1)*1

= 6 + (2*1)

= 8

Hence, the 3rd term is 8
 
Answer.