# Find the sum of n terms of the series 1.3.5 + 3.5.7 + 5.7.9 + ……

## Solution :

By using method of differences,

The $$n^{th}$$ term is (2n-1)(2n+1)(2n+3)

$$T_n$$ = (2n-1)(2n+1)(2n+3)

$$T_n$$ = $$1\over 8$$(2n-1)(2n+1)(2n+3){(2n+5) – (2n-3)}

= $$1\over 8$$($$V_n$$ – $$V_{n-1}$$)

$$S_n$$ = $${\sum}_{r=1}^{n‎} T_n$$ = $$1\over 8$$($$V_n$$ – $$V_0$$)

$$\therefore$$  $$S_n$$ = $$(2n-1)(2n+1)(2n+3)(2n+5)\over 8$$ + $$15\over 8$$

= $$n(2n^3 + 8n^2 + 7n – 2)$$

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