Question

An arithmetic sequence has as its first three terms `a_1=a, a_2=2a, a_3=a^2`. What are the three terms?

Answer 1

See below:

Explanation:

The formula for an arithmetic sequence is:

`a_n=a_1+(n-1)d` where:

• `a_n` is the `n^("th")` term
• `a_1` is the first term
• `n` is the numbered term in the sequence
• `d` is the amount by which the terms increase (or decrease) by.

If I have a sequence that starts with `a_1=0` and `d=2`, I'll have:

`a_1=0+(1-1)2=0`
`a_2=0+(2-1)2=2`
`a_3=0+(3-1)2=4`
`a_4=0+(4-1)2=6`

and so on - we end up listing the set of positive even numbers with this sequence.

In our question, we have as terms `a_1=a, a_2=2a, a_3=a^2`

Let's plug what we know into our general formula for the first two terms:

`a_1=a+(1-1)d=a+0d=a`
`a_2=a+(2-1)d=a+d=2a`

We now know that `d=a`

`a_3=a+(3-1)a=a+2a=3a=a^2`

Here we can now see that there are only 2 values of `a` that will satisfy the equation: `a=0,3`. We want the non-zero value and so `a=3`.

The sequence then is `a, 2a, a^2 = 3, 2(3), 3^2=3, 6, 9`