sum of five middle most terms is 195 and no. of terms is 43 find the sum of AP
Answers
Answer: 1677
Step-by-step explanation:
Sum of 5 middlemost terms = 195
No. of terms in A.P. = 43
Let’s assume the terms in A.P. be as follows:
a₁ = a, a₂ = a + d, a₃ = a + 2d, a₄= a + 3d, .......
The middle term in AP is given as,
= (n+1)/2
= (43 + 1)/2
= 22
∴ The 5 middle terms are: a₂₀, a₂₁, a₂₂, a₂₃ & a₂₄
According to the question, we have
a₂₀ + a₂₁ + a₂₂ + a₂₃ + a₂₄ = 195
⇒ (a + 19d) + (a + 20d) + (a + 21d) + (a + 22d) + (a + 23d) = 195
⇒ 5a + 19d + 20d + 21d + 22d + 23d = 195
⇒ 5a + 105 d = 195
⇒ a + 21 d = 39 ……. (i)
Thus, the sum of the 43 terms in A.P. i.e.,
= a₁+ a₂ + a₃ …….. + a₄₃
= a + (a+d) + (a+2d) + ....... + (a+ 42d)
= 43a + [(42*43)/2]d …….. [since sum of n terms in A.P. is Sₙ= n/2{2a+(n-1)d}]
= 43a + 903d
= 43 ( a + 21d)
On substituting from eq. (i) we get,
= 43 x 39
= 1677