Question

Develop an explicit formula in terms of n for the nth term of the following sequence:

0,4,18,48,100,180,......

show that it "works" for the sixth term and use it to find the seventh term.

Answer 1

Answer with explanation:

The given sequence : 0,4,18,48,100,180,......

We can write the terms of the sequence as :

`\text{Ist term }:a_1=1(1^2-1)=0\\\\\text{IInd term }: a_2=2^2(2-1)=4(2-1)=4\\\\\text{IIIrd term }:a_3=3^2(3-1)=9(2)=18\\\\\text{IVth term }:a_4=4^2(4-1)=(16)(3)=48\\\\\text{Vth term }:a_5=5^2(5-1)=25(4)=100\\\\\text{VIth term }:a_6=6^2(6-1)=36(5)=180`

From the above presentation of the terms, the explicit formula in terms of n for the nth term will be :-

`a_n=n^2(n-1)`

Put n= 7 , we get

`a_7=7^2(7-1)=49(6)=294`

Therefore, the seventh term of the given sequence = 294