In an AP the sum of
second & third term is 22 & the
Product of first & fourth term is
85. Then find the sum of first
10 terms by considering the
positive value of common
difference.
Answers
Answer:
When the consecutive terms of series differ by a common number, then the series is said to be Arithmetic Progression
Let a be the first term of the AP
d be the common difference of the AP
nth term of AP ⇒ a + ( n - 1) d
Given,
The sum of second and third term is 22
⇒ (a + d) + (a + 2d) =22
⇒ 2a + 3d = 22
⇒ d = 1/3 ( 22 - 2a)
The product of first and fourth term is 85
⇒ a ( a + 3d) = 85
⇒ a² + 3ad = 85
Substituting the value of d gives,
⇒ a ( a + 3 ( 1/3 * (22 - 2a) )) = 85
⇒ a ( a + 22 - 2a) = 85
⇒ a ( - a + 22) = 85
⇒ - a² + 22a = 85
⇒ a² - 22a + 85 = 0
⇒ a² - 17a - 5a + 85 = 0
⇒ a ( a - 17) - 5 ( a - 17)= 0
⇒ (a-5)(a-17)= 0
⇒ a = 5 or a = 17
If a = 5,
d = 1/3 ( 22 - 10) = 1/3 ( 12) = 4
If a = 17,
d = 1/3 ( 22 - 34) = 1/3 ( - 12) = - 4
a = 5, d = 4
Then Arithmetic Progression is
5, 9, 13, 17
a = 17, d = - 4
Then Arithmetic Progression is,
17, 13, 9, 5
Therefore, The required terms in the AP are 5, 9, 13, 17.