Pay close attention to how the terms are related and how each term is related to its position.
∑^(∞)_(n=0) (i^2+i)
Practice makes perfect
We want to write the given series in summation notation.
0+2+6+12+⋯
If we write this series as a list of numbers and consider the ratios and differences between them, we can can conclude that it is neither an arithmetic sequence nor a geometric sequence.
0, 2, 6, 12, ⋯
Therefore, we need to try a different approach. Let's think of 0 as the 0 th term. Then 2 will be the 1 st term and so on. Keeping this in mind, let's rewrite each term as a sum.
0, 2, 6, 12, ⋯
⇕
0+ 0, 1 + 1, 4 + 2, 9+ 3 , ...
Now, notice that the first parts of these sums are square numbers. We can use this and the pattern we came up with above to write an explicit formula for our sequence.
a_0
a_1
a_2
a_3
...
a_i
0^2+ 0
1^2+ 1
2^2+ 2
3^2+ 3
...
i^2+ i
0+0
1+1
4+2
9+3
...
We found the explicit formula for our sequence.
a_i = i^2+i
Note that the given series is infinite. Therefore, the upper limit of the series is ∞. Since we decided that 0 is the 0 th term, the sum will begin with i= 0.
∑^(∞)_(i= 0) (i^2+i)
