The sum of the 4th and 8th term of an A.P is 24 and the sum of the 6th and 10th term is 34 .find the first four term and the 20th term of the A.P
Answers
given :-
sum of 4th and 8th term of an AP = 24
and sum of the 6th and 10th term = 34
we know that,
a1 = a , a2 = a + d, a3 = a + 2d
similarly,
a4 = a + 3d, a6 = a + 5d, a8 = a + 7d and a10 = a + 9d
ATQ,
a4 + a8 = 24
➡ a + 3d + a + 7d = 24
➡ 2a + 10d = 24
➡ 2a + 10d - 24 = 0 --------(i)
and
a6 + a10 = 34
➡ a + 5d + a + 9d = 34
➡ 2a + 14d = 34
➡ 2a + 14d - 34 = 0 --------(ii)
using elimination method,
subtracting equation (ii) from equation (i)
➡ (2a + 10d - 24) - (2a + 14d - 34) = 0
➡ 2a + 10d - 24 - 2a - 14d + 34 = 0
➡ 2a - 2a + 10d - 14d - 24 + 34
➡ -4d + 10 = 0
➡ -4d = -10
➡ d = -10/-4 = 5/2
d = 5/2
now putting value of d in equation (i)
➡ 2a + 10(5/2) = 24
➡ 2a + 50/2 = 24
➡ 2a = 24/1 - 50/2
➡ 2a = (48 - 50)/2
➡ 2a = -2/2 = -1
➡ a = -1/2
a = -1/2
now, we've found the values of a and d.
let's find the first four terms of the AP.
a1 = a = -1/2
a2 = a1 + d = -1/2 + 5/2 = 4/2 = 2
a3 = a2 + d = 2 + 5/2 = 9/2
a4 = a3 + d = 9/2 + 5/2 = 14/2 = 7
hence, the first four terms of the AP are -1/2, 2, 9/2 and 7
20th term of AP = a + 19d
= -1/2 + 19(5/2)
= -1/2 + 95/2
= 94/2 = 47
According to question,
a4 + a8 = 24
a + 3d + a + 7d = 24
2a + 10d = 24
2a + 10d - 24 = 0 ....(1)
also
a6 + a10 = 34
a + 5d + a + 9d = 34
2a + 14d = 34
2a + 14d - 34 = 0 ...(2)
on solving (1) & (2) we get;
d = 10/4 = 5/2
put value of d in (1)
2a + 10(5/2) = 24
2a + 50/2 = 24
2a = 24/1 - 50/2
2a = (48 - 50)/2
2a = -2/2 = -1
a = -1/2
a = -1/2
AP....
a1 = a = -1/2
a2 = a1 + d = -1/2 + 5/2 = 4/2 = 2
a3 = a2 + d = 2 + 5/2 = 9/2
a4 = a3 + d = 9/2 + 5/2 = 14/2 = 7
20th term of an AP
= a + 19d
= -1/2 + 19(5/2)
= -1/2 + 95/2
= 94/2 = 47