# How to Use Summation Notation to Show a Partial Sum of a Sequence

Summation notation is a useful way to represent the partial sum of a sequence. The sum of the first *k* terms of an arithmetic sequence is referred to as the* k*th partial sum*.* They're called partial sums because you're only able to find the sum of a certain number of terms — no infinite series here! You may use partial sums when you want to find the area under a curve (graph) between two certain values of *x.* Although finding the *entire* area under the graph isn't always possible (because it could be infinite if the curve goes on forever), you can find the area underneath a piece of it.

Don't let the use of the *k* variable confuse you. Instead of *k, *your book may use *n* and call it an *n*th partial sum. Remember that a variable just stands in for an unknown, so it really can be any variable you want — even a Greek variable. But most books use *k* to represent the number of terms in a series and *n* for the number of terms in a sequence.

The notation of the *k*th partial sum of a sequence is as follows:

You read this equation as "the *k*th partial sum of *a*_{n}* *is . . ." where *n* = 1 is the *lower limit *of the sum and *k* is the *upper limit *of the sum. To find the *k*th partial sum, you begin by plugging the lower limit into the general formula and continue in order, plugging in integers until you reach the upper limit of the sum. At that point, you simply add all the terms to find the sum.

To find the fifth partial sum of *a*_{n}* *= *n*^{3} – 4*n* + 2, for example, follow these steps:

Plug all values of

*n*(starting with 1 and ending with*k*) into the formula.Because you want to find the fifth partial sum, plug in 1, 2, 3, 4, and 5:

*a*_{1}= (1)^{3}– 4(1) + 2 = 1 – 4 + 2 = –1*a*_{2}= (2)^{3}– 4(2) + 2 = 8 – 8 + 2 = 2*a*_{3}= (3)^{3}– 4(3) + 2 = 27 – 12 + 2 = 17*a*_{4}= (4)^{3}– 4(4) + 2 = 64 – 16 + 2 = 50*a*_{5}= (5)^{3}– 4(5) + 2 = 125 – 20 + 2 = 107

Add all the values from

*a*_{1}to*a*to find the sum._{k}This step gives you

–1 + 2 + 17 + 50 + 107 = 175

Rewrite the final answer, using summation notation.