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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

Theodore Amidon

See below:

The formula for an arithmetic sequence is:

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

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#

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Answer 2

Caleb Alexander

The three terms of the arithmetic sequence are ( a ), ( 2a ), and ( a^2 ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.