if the sum of first 7 term of an ap is 49 and that of n term is 289 . find out sum of first n term
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Step-by-step explanation:
Sum of the first n terms of an AP is given by Sₙ = n/2 [2a + (n - 1) d] or Sₙ = n/2 [a + l], and the nth term of an AP is aₙ = a + (n - 1)d
Here, a is the first term, d is the common difference and n is the number of terms and l is the last term.
Given,
Sum of first 7 terms, S₇ = 49
Sum of first 17 terms, S₁₇ = 289
We know that sum of n terms of AP is Sₙ = n/2 [2a + (n - 1) d]
S₇ = 7/2 [2a + (7 - 1)d]
49 = 7/2 [2a + 6d]
a + 3d = 7 ... (i)
S₁₇ = 17/2 [2a + (17 - 1) d]
289 = 17/2 [2a + 16d]
a + 8d = 17 ... (ii)
Subtracting equation (i) from equation (ii),
a + 8d - (a + 3d) = 17 - 7
5d = 10
d = 2
From equation (i),
7 = a + 3 × 2
7 = a + 6
a = 1
Sₙ = n/2 [2a + (n - 1) d]
= n/2 [2 × 1 + (n - 1) 2]
= n/2 [2 + 2n - 2]
= n/2 × 2n
= n2
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