find the sum of the following series to n terms: (i) 3+5+9+15+23_ _ _ - Maths - Sequences and Series

Question

find the sum of the following series to n terms: (i)
3+5+9+15+23_ _ _

Santosh Kumar · Accepted Answer

3 + 5 + 9 + 15 + 23 + The sequence of the differences between the successive terms is 2, 4, 6, 8, , which is an A P $S_{n} = 3 + 5 + 9 + 15 + 23 + (1) S_{n} = 3 + 5 + 9 + 15 + 23 + (2)$ Subtracting (2) from (1), $0 = 3 + [2 + 4 + 6 + 8 + (n - 1) terms] - a_{n} \Rightarrow 3 + \frac{(n - 1)}{2} [2 (2) + (n - 2) 2] - a_{n} = 0 \Rightarrow 3 + (n - 1) n - a_{n} = 0 \Rightarrow a_{n} = n^{2} - n + 3 Required sum = Σ_{r = 1}^{n} r^{2} - r + 3 = Σ_{r = 1}^{n} r^{2} - Σ_{r = 1}^{n} r + Σ_{r = 1}^{n} 3 = \frac{n (n + 1) (2 n + 1)}{6} - \frac{n (n + 1)}{2} + 3 n = \frac{n (n + 1) (2 n + 1) - 3 n (n + 1) + 18 n}{6} = \frac{n (n^{2} + 8)}{3}$