Adding up the terms of a sequence from Recursive Formulas is so common that it has its own symbol, the Greek capital letter sigma, $\Sigma$, for "sum."
Key idea
$$\sum_{k=m}^{n} f(k) = f(m) + f(m + 1) + \cdots + f(n)$$ Plug in each whole number $k$ from $m$ to $n$ and add the results. The number of terms is $n - m + 1$.
Why it works
Sigma notation is a compact recipe: the expression after $\Sigma$ makes each term, and the numbers below and above say where the index starts and stops. Counting the terms: from $m$ to $n$ there are $n - m$ steps, but both ends count, so there are $n - m + 1$ terms, just as there are 3 numbers from 5 to 7.
Worked examples
Worked example: Evaluating a sum
Evaluate $\displaystyle\sum_{k=2}^{5} (3k - 1)$.
- $k = 2, 3, 4, 5$ give $5, 8, 11, 14$.
- Add: $5 + 8 + 11 + 14 = 38$.
Answer: $38$
Worked example: Counting terms
How many terms are in $\displaystyle\sum_{k=4}^{20} k^2$?
- $20 - 4 + 1 = 17$.
Answer: 17 terms
Worked example: Writing a sum in sigma notation
Write $3 + 7 + 11 + 15 + 19$ in sigma notation.
- The terms go up by 4, so the general term is $4k + c$. With $k = 1$ the term is 3, so $c = -1$.
- There are 5 terms, so $k$ runs from 1 to 5. Check: $4(5) - 1 = 19$.
Answer: $\displaystyle\sum_{k=1}^{5} (4k - 1)$
Two handy facts: a constant added $N$ times is $N$ times the constant, so $\sum_{k=1}^{10} 4 = 40$; and sums split, so $\sum (a_k + b_k) = \sum a_k + \sum b_k$ and $\sum c\,a_k = c\sum a_k$. They follow from regrouping and factoring an ordinary sum.
Common mistake
Miscounting terms when the index doesn't start at 1. $\sum_{k=4}^{20}$ has 17 terms, not 16 and not 20. Another slip is starting the substitution at $k = 1$ out of habit. Always begin with the number written under the sigma.