Recipe

Write an explicit sum in sigma notation

Use when: you have a sum written out term by term and need to compress it into one \sum expression

Goal: a single \sum expression — index letter, starting value, ending value, and general term — that expands back out to the exact sum you started with

Taught in Sigma notation and index bookkeeping

1. 1

   Line the terms up and compare them. Everything that looks identical from term to term is fixed; the one piece that's different in each term — almost always a subscript — is what the index will track.

   Example

   In a_1b_1 + a_2b_2 + a_3b_3 + a_4b_4, every term has the same shape, a_{\square}b_{\square}, with a matching pair of subscripts multiplied together. The letters a and b, and the multiplication between them, never change; only the subscript does.

   Why: the index is only allowed to replace the ONE thing that's actually changing — treating a fixed piece as if it changed produces a general term that stops matching the real terms after the first one.

2. 2

   Name the index (commonly i, j, or k), then write a formula in that index that reproduces the changing piece exactly as the index counts up by 1 each term.

   Example

   The subscript in a_1b_1+a_2b_2+a_3b_3+a_4b_4 goes 1, 2, 3, 4 — counting up by exactly 1 each term — so the index itself, with no adjustment, already is the changing piece: general term a_ib_i.

   • the changing piece counts by 1 each term (1,2,3,\ldots) — the index itself is the general term's changing piece, with no formula needed on top of it.
   • the changing piece skips or scales (e.g. only odd numbers 1,3,5,\ldots, or even numbers 2,4,6,\ldots) — build a small formula in the index, like 2i-1 or 2i, that hits those exact values as i itself counts 1,2,3,\ldots

   Why: whichever case applies, this step is what turns "a pattern I can see" into "a formula I can substitute into," which is the only thing a sigma expression is actually allowed to contain.

3. 3

   The first term you wrote tells you where the index starts; the last term tells you where it stops.

   Example

   The first term, a_1b_1, has subscript 1; the last term, a_4b_4, has subscript 4 — so i runs from 1 to 4.

   Why: a sigma expression with a general term but no stated bounds doesn't tell a reader how many terms to add, or where to stop.

4. 4

   Combine the index, its starting and ending values, and the general term into one \sum expression.

   Example

   a_1b_1+a_2b_2+a_3b_3+a_4b_4 = \sum_{i=1}^{4} a_ib_i

   Why: this is the single compact object the previous three steps existed to build — the index, its range, and the formula driving it, all in one symbol.

5. 5

   Substitute the starting value, then every value up to the ending value, back into the general term, and check that the resulting list of terms matches the original sum exactly, in the same order.

   Example

   Expanding \sum_{i=1}^{4}a_ib_i back out gives a_1b_1+a_2b_2+a_3b_3+a_4b_4 — exactly the sum this recipe started from, confirming the sigma expression is correct.

   Why: a general term and bounds can look plausible and still be wrong — expanding back out is the only step that actually confirms they reproduce every original term, not just the first and last.