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Find the nth term of the given arithmetico-geometric series:(i) \(\frac{1}{3}\) + \(\frac39\) + \(\frac5{27}\) + \(\frac{7}{81}\) + ............(ii) 1 – 2x + 3x2 – 4x3 + ..... |
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Answer» (i) The A.P. and G.P. corresponding to the given series \(\frac{1}{3}\) + \(\frac39\) + \(\frac5{27}\) + \(\frac{7}{81}\) + ............ are respectively 1, 3, 5, 7, .............. and \(\frac{1}{3}\), \(\frac39\), \(\frac5{27}\), \(\frac{7}{81}\),........... nth term of A.P. = (1 + (n – 1)2) = (2n – 1) nth term of G.P. = \(\frac{1}{3}\). \(\big(\frac{1}{3}\big)^{n-1}\) = \(\big(\frac{1}{3}\big)^{n}\) ∴ nth term of the given series = (2n - 1)\(\big(\frac{1}{3}\big)^{n}\). (ii) The given arithmetico-geometric series is 1 – 2x + 3x2 – 4x3 + ..... where corresponding A.P. and G.P. are respectively 1, 2, 3, 4, .... and 1, – x, (– x)2, (– x)3, ..... nth term of A.P = (1 + (n – 1) 1) = n nth term of G.P = 1 . (– x)n – 1 = (– 1)n – 1 (x)n – 1. ∴ nth term of the given series = n . (– 1)n – 1 xn – 1 = (– 1)n – 1 nxn – 1. |
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