Index of summation?

Question

Using summation notation to rewrite the following sum with the index of summation starting at the indicated number.

(1) 6
SigmaE (5+3i) ;3(indicated number)
i=1

(2) 6
E (i^3-3) ; 0(indicated number)
i=1

Thanks in advance for guidance and help!

Answer 1

(1)
6
Σ (5+3i) =
i=1

8
Σ (5+3(i - 2)) (i from 2 later → i in the term must decrease by 2) =
i=3

8
Σ (3i - 1))
i=3



(2)
6
Σ (i^3-3) (i from 1 earlier → i in the term must increase by 1) =
i=1

5
Σ ((i+1)^3-3)
i=0

Answer 2

1. Since there are six terms on the original, you must still have six terms, so change the top index to 8 (you add 2 to it since you increased the bottom index by 2).
Then change the i to (i-2) so that when i starts at 3, you get 3-2 which is what it would have been originally.
(5 + 3(i-2)) = 5 + 3i - 6 = -1 + 3i

so the formula is now that.

2. Decrease the 6 on top to 5 (since the bottom index got decreased by 1); replace i with (i+1) so that when you start at i=0 wou get what it would have been to start originally.

so it's now (i+1)^3 - 3

Answer 3

Instead of summing from 1 to 6, we want to start at 3. So its going to be from 3 to 8, with everything shifted up by 2. So, to counter for that, we need to shift i down by 2 in the actual formula: so you get (sum from i=3 to 8) 5+3(i-2), or 3i-1.

Similarly, for the second one, we shift the index down by 1 in the sum, so we want to shift i up by 1 in the formula. So we get (sum from i=0 to 5) (i+1)^3 - 3 (which you could expand if you wanted to.)