The sum of 4th and 8th term of an ap is 24 and the sum of the 6th and 10th term is 34 find the first term and the common difference of the AP.
Answers
Answer:-
Given:
Sum of 4th and 8th terms of an AP = 24
We know that,
nth term of an AP = a + (n - 1)d
Hence,
→ a(4) + a(8) = 24
→ a + (4 - 1)d + a + (8 - 1)d = 24
→ 2a + 3d + 7d = 24
→ 2a + 10d = 24
→ 2(a + 5d) = 24
→ a + 5d = 24/2
→ a + 5d = 12 -- equation (1)
Similarly,
→ a(6) + a(10) = 34
→ a + 5d + a + 9d = 34
→ 2a + 14d = 34
→ 2(a + 7d) = 34
→ a + 7d = 34/2
→ a + 7d = 17 -- equation (2)
Subtracting equation (1) from (2).
→ a + 7d - (a + 5d) = 17 - 12
→ a + 7d - a - 5d = 5
→ 2d = 5
→ d = 5/2
Putting the value of d in equation (1) we get,
→ a + 5d = 12
→ a + 5(5 / 2) = 12
→ a + 25/2 = 12
→ a = 12 - 25/2
→ a = (24 - 25) / 2
→ a = - 1/2
Hence, the first term and the common difference of the given AP are 5/2 and - 1/2.
Answer:
First term is -1/2 and common difference is 5/2.
Step-by-step explanation:
The sum of 4th and 8th term of an AP is 24.
→ a + (4-1)d + a + (8-1)d = 24
→ a + 3d + a + 7d = 24
→ 2a + 10d = 24
→ 2( a + 5d ) = 24
→ a + 5d = 12
→ a = 12 - 5d
Sum of the 6th and 10th term is 34.
→ a + (6-1)d + a + (10-1)d = 34
→ a + 5d + a + 9d = 34
→ 2a + 14d = 34
On taking 2 as common we get,
→ a + 7d = 17
→ a = 17 - 7d
On comparing both equations we get,
→ 12 - 5d = 17 - 7d
→ 12 - 17 = -7d + 5d
→ 5 = 2d
→ d = 5/2
So,
→ a = 12 - 5d
→ a = 12 - 5 × (5/2)
→ a = 12 - 25/2
→ a = (24 - 25)/2
→ a = -1/2