find the sum of 51 terms of the AP whose second term is 2 and the 4th term is 8
Answers
EXPLANATION.
Sum of 51 terms of an A.P.
second term = 2.
Fourth term = 8.
As we know that,
General term of an A.P.
⇒ Tₙ = a + (n - 1)d.
⇒ T₂ = a + (2 - 1)d.
⇒ T₂ = a + d.
⇒ a + d = 2. - - - - - (1).
⇒ T₄ = a + (4 - 1)d.
⇒ T₄ = a + 3d.
⇒ a + 3d = 8. - - - - - (2).
From equation (1) and (2), we get.
Subtract equation (1) and (2), we get.
⇒ a + d = 2. - - - - - (1).
⇒ a + 3d = 8. - - - - - (2).
⇒ - - -
We get,
⇒ - 2d = - 6.
⇒ 2d = 6.
⇒ d = 3.
Put the value of d = 3 in the equation (1), we get.
⇒ a + d = 2.
⇒ a + 3 = 2.
⇒ a = 2 - 3.
a = - 1.
First term = a = - 1.
Common difference = d = 3.
As we know that,
Sum of nth term of an A.P.
⇒ Sₙ = n/2[2a + (n - 1)d].
⇒ S₅₁ = 51/2[2(-1) + (51 - 1)3].
⇒ S₅₁ = 51/2[-2 + 50 x 3].
⇒ S₅₁ = 51/2[-2 + 150].
⇒ S₅₁ = 51/2[148].
⇒ S₅₁ = 51 x 74.
⇒ S₅₁ = 3774.
MORE INFORMATION.
Supposition of term in an A.P.
(1) = Three terms as : a - d, a, a + d.
(2) = Four terms as : a - 3d, a - d, a + d, a + 3d.
(3) = Five terms as : a - 2d, a - d, a, a + d, a + 2d.
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How To Find the Sum of Arithmetic Progression?
To find the sum of arithmetic progression, we have to know the first term, the number of terms, and the common difference between consecutive terms. Then the formula to find the sum of an arithmetic progression is Sn = n/2[2a + (n − 1) × d] where, a = first term of arithmetic progression, n = number of terms in the arithmetic progression and d = common difference.
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