# What is the value of the sum of the sequence: #1+5+14+30+...........# upto n terms? Thank you!

##### 1 Answer

#### Explanation:

Notice that the sequence of differences between terms follows the pattern

Assuming the pattern continues, the next couple of terms would be

Write down the sequence of the first few sums:

#color(blue)(1), 6, 20, 50, 105, 196#

Write down the sequence of differences between consecutive terms:

#color(blue)(5), 14, 30, 55, 91#

Write down the sequence of differences of that sequence:

#color(blue)(9), 16, 25, 36#

Write down the sequence of differences of that sequence:

#color(blue)(7), 9, 11#

Write down the sequence of differences of that sequence:

#color(blue)(2), 2#

Having reached a constant sequence, we can use the initial terms of each of these sequences as coefficients of an expression for the

#s_n = color(blue)(1)/(0!) + color(blue)(5)/(1!)(n-1) + color(blue)(9)/(2!)(n-1)(n-2) + color(blue)(7)/(3!)(n-1)(n-2)(n-3) + color(blue)(2)/(4!)(n-1)(n-2)(n-3)(n-4)#

#color(white)(s_n) = 1+5(n-1)+9/2(n^2-3n+2)+7/6(n^3-6n^2+11n-6)+1/12(n^4 - 10n^3 + 35n^2 - 50n + 24)#

#color(white)(s_n) = 1/12(n^4 + 4n^3 + 5n^2 + 2n)#