High School Math Solutions – Algebra Calculator, Sequences

When dealing with simpler sequences, we can look at the sequence and get a feel for what the next term or the rule is.  However, not all sequences are nice like those. Finding the rule of a sequence can be difficult if you don’t know where to begin. We will focus on arithmetic and geometric sequences, which will make finding the rule, term, and sum of terms very easy.

Arithmetic Sequence:

• A sequence where the difference between the consecutive terms is constant
• Finding the nth term: a_n=a_1+(n-1)d
• Find the sum of n terms: S_n=\frac{n(a_1+a_n)}{2}

Geometric Sequence:

• A sequence where each term is multiplied the previous term by a constant
• Finding the nth term: a_n=a_0\cdot r^{n-1}
• Find the sum of n terms: S_n=\frac{a_1(1-r^n)}{1-r}

Here we will go through a problem involving an arithmetic sequence step by step (click here):

1. See if you can see a pattern

Find the next 3 terms of the following sequence: 3,11,19,27,35,...
We can see that difference in each term is 8. 

2. Plug terms into the formula

a_n=a_1+(n-1)d
a_n=3+(n-1)8

3. Compute terms

a_6=3+(6-1)8=43
a_7=3+(7-1)8=51
a_8=3+(8-1)8=59

We will now go through a problem involving a geometric sequence step by step (click here):

Find the 7th term of the following sequence:  \frac{2}{9},\frac{2}{7},2,6,18

1. Check to see if the ratio is constant

2. Plug terms into formula

a_n=a_0∙r^(n-1)
a_n=\frac{2}{9}\cdot 3^(n-1)

3. Compute term

a_7=\frac{2}{9}\cdot 3^6=162

Here’s an example where you have to find the sum of terms (click here):

Arithmetic and geometric series aren’t very hard. Just memorize the formulas and it should be pretty easy.

Until next time,
Leah