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प्रश्न
Find Sn of the following arithmetico - geometric sequence:
1, 4x, 7x2, 10x3, 13x4, …
उत्तर
1, 4x, 7x2, 10x3, 13x4, …
Here, 1, 4, 7, 10, 13, … are in A.P.
∴ a = 1, d = 3
∴ nth term = a + (n – 1)d
= 1 + (n – 1)(3)
= 3n – 2
1, x, x2, x3, … are in G.P.
∴ a = 1, r = x,
∴ nth term = arn–1 = xn–1
∴ nth term of arithmetico-geometric sequence is
tn = (3n – 2)xn–1
∴ Sn = 1 + 4x + 7x2 + 10x3 + …+ (3n – 2)xn–1 ...(i)
Multiplying throughout by x, we get
xSn = x + 4x2 + 7x3 + 10x4 + …+ (3n – 2)xn ...(ii)
Equation (i) – equation (ii), we get
Sn – xSn = 1+ 3x + 3x2 + 3x3 + … + 3xn–1 – (3n – 2).xn
∴ (1 – x)Sn = `1 + 3x ((1 - x^("n" -1))/(1 - x)) - (3"n" - 2).x^"n"`
∴ (1 – x)Sn = `1 - (3"n" - 2).x^"n" + (3x(1 - x^("n" -1)))/(1 - x)`
∴ Sn = `(1 - (3"n" - 2)x^"n")/(1 - x) + (3x(1 -x^("n" -1)))/(1 - x)^2`
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