Formula nth term series 4,9,16,25

Question

Hello, could you help me find the formula for this series, please? How do I go about it? 

4,9,16,25,...I know it starts with adding 5 and adding 2 more with every number, so adding 7 next, then 9, then 11...it is an arithmetic progression, no? As it's adding up. But how do I use the formula a+(n-1)d?I have a=4 but I can't set d, as it changes. 

Answer 1

You can't use the formula a+(n-1)d for exactly the reason that you give (d changes).  The formula is ONLY for arithmetic sequences where d remains constant.

 

The sequence 4, 9, 16, 25, ... is not arithmetic, but 4, 9, 16, 25,... are perfect squares.  

 

a₁ = 4 = 2²

a₂ = 9 = 3²

a₃ = 16 = 4², etc

 

So, we could define the sequence as a_n = (n+1)², for n = 1,2,3,... 

 

 

Answer 2

Hi Xenia,

 

If you look at the first 4 terms given, you'll see that they are squares of 2,3,4,5.

 

Notice that 1² or 1 is not part of the series

The first term is 2², the 2nd term is 3², etc.  So the nth term would be (n+1)²

 

So the formula for the nth term is

 

(n+1)²