Sum of the First $$n$$ Terms

(of an Arithmetic Sequence)

Given an arithmetic sequence we'll sometimes need to calculate the sum of its first $$n$$ terms.

For example, given the arithmetic sequence whose first few terms are: $3,7,11,15,19,23, \dots$ we may need to calculate the sum of its first $$100$$ terms.
We could do this by adding one term to the next up to the $$100^{\text{th}}$$ term but that would take time.
Instead we use one of two formula.

We have two formula for the sum of the first $$n$$ terms of an arithmetic sequence, we learn both in this section.

Formula 1

Given an arithmetic sequence, we can calculathe the sum of its first $$n$$ terms, which we write $$S_n$$, using the formula: $S_n = \frac{n}{2} \begin{pmatrix} u_1 + u_n \end{pmatrix}$ Where $$u_1$$ is the first term of the sequence and $$u_n$$ is the $$n^{\text{th}}$$ term.

So, for example, $$S_{10}$$ refers to the sum of the first $$10$$ terms, $$S_{250}$$ refers to the sum of the first $$250$$ terms, ... .

The method for using this formula is illustrated in the tutorial below.

Tutorial

In the following tutorial we review the method for findng the formula for the $$n^{\text{th}}$$ term of a linear sequence. Watch it now.

Exercise

1. Calculate the sum of the first $$20$$ terms of the arithmetic sequence whose formula for the $$n^{\text{th}}$$ term is: $u_n = 1 + (n-1)\times 4$

2. Calculate the sum of the first $$30$$ terms of the arithmetic sequence defined by the formula: $u_n = 4+(n-1)\times (-3)$

3. Calculate the sum of the first $$25$$ terms of the arithmetic sequence defined by the formula: $u_n = 5+ (n-1)\times 7$

4. Given an arithmetic sequence defined by the formula: $u_n = 11 + (n-1)\times 6$ Calculate the sum of the $$6^{th}$$ to the $$20^{\text{th}}$$ terms included.
Hint: this equals to the difference of two sums.

Formula 2

Given an arithmetic sequence, we can calculate the sum of its first $$n$$ terms, $$S_n$$, using the formula: $S_n = \frac{n}{2} \begin{pmatrix} 2.u_1 + (n-1)d \end{pmatrix}$ Where $$u_1$$ is the first term of the sequence and $$d$$ is its common difference.

Note: exam questions frequently involve this second formula.

Exercise 2

1. Given the sequence whose first few terms are: $-2,3,8,13,18, \dots$ calculate the sum of its first $$20$$ terms.

2. Given the sequence whose first few terms are: $7,2,-3,-8,-13, \dots$ calculate the sum of its first $$100$$ terms.

3. Given the sequence whose first few terms are: $1,4,7,10,13, \dots$ calculate the sum of its first $$50$$ terms.

4. Given the sequence whose first few terms are: $0,-3,-6,-9,-12, \dots$ calculate the sum of its first $$200$$ terms.

Must Know Exercises

We now learn how to solve some tyical exam-type questions involving the sum of the first $$n$$ terms of an arithmetic sequence.

Exercise

1. An arithmetic sequence has eigth term equal to $$33$$ and the sum of its first fifteen terms is $$660$$.
Find the values of the sequence's first term, $$u_1$$, and of its common difference, $$d$$.

2. The sum of the first $$20$$ terms of an arithmetic sequence is $$550$$.
Given it has first term equal to $$-2$$, find the value of the common difference $$d$$.