Question

Example
Find the sum to n-terms: 2 + 12 + 36 + 80 + 156 + 252 + 392
....

Answer 1

Answer:

What is the next number in the series 2, 12, 36, 80, and 150?

Ans. Series is 2, 12, 36, 80, 150, ?

Step -I Difference b/w steps of given series -

10, 24, 44, 70, X (let diff b/w 150 & ? is X)

Step -II Difference b/w series of steps I-

14, 20, 26, Y (let diff b/w 70 & X is Y)

Step III Difference b/w series of Step II-

6, 6, 6 ( Here 6 is common diff b/w all steps)

Now,

Step II will be - 14, 20, 26, 32 ( Because Y = 26+6)

Step I will be- 10, 24, 44, 70, 102 ( Because X = 70+32)

therefore given series will be-

2, 12, 36, 80, 150, 252 ( Because ? = 150+102)

So answer is 252.

Answer 2

Let S = 2 + 12 + 36 + 80 + 150 + 252 + ................+Tn ...........(i)  

S = 2 + 12 + 36 + 80 + 150 + 252 + .........+Tn–1 + Tn ...........(ii)

(i) – (ii) ⇒ Tn = 2 + 10 + 24 + 44 + 70 + 102 + ............... + (Tn – Tn–1) ...........(iii)  

Tn = 2 + 10 + 24 + 44 + 70 + 102 + ....... + (Tn–1–Tn–2) + (Tn – Tn–1) ...........(iv)  

(iii) – (iv) ⇒ Tn – Tn–1 = 2 + 8 + 14 + 20 + 26 + .........  

= n/2[4 + (n – 1) 6] = n [3n – 1]

= Tn – Tn–1

= 3n2 – n

∴ general term of given series is ∑ Tn – Tn–1 = ∑ 3n2 – n = n3 + n2.  Hence sum of this series is S = ∑ n3 + ∑ n2