In an A.P the sum of second and third term is 22, and the product of first and
fourth term is 85. Find the first four terms.
Answers
Answer:
In an A.P the sum of second and third term is 22, and the product of first and
fourth term is 85. Find the first four terms.
The required terms in the AP are 5, 9, 13, 17.
Step-by-step explanation:
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:
S (2nd term + 3rd term) is 22
⇒ (a + d) + (a + 2d) =22
⇒ 2a + 3d = 22
⇒ d = (22- 2a) × 1/3
The product of the 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.