Question

Need help with problem #53! Arithmetic sequences!

Answer 1

When counting any sequence, it helps to have a simpler sequence to compare it to. The simplest one that I can think of is

$1,2,3,4,5,\dots$

because you instantly can tell the number of terms in the sequence by looking at the last number. We can see from the graph that the first few terms of the sequence a are 1, -3, -7, and we're told that its last term is -83. Our goal then is to turn this sequence:

$1,-3,-7,\dots,-83$

into this one:

$1,2,3,\dots$

The first thing which stands out in this sequence is the number of negative terms, so let's fix that by multiplying every term by -1:

$1,-3,-7,\dots,-83\xrightarrow{\times{-1}}-1,3,7,\dots,83$

Now, the main property of any arithmetic sequence is that they increase or decrease by some constant amount. Here, that number is 4, since -3 = 1 - 4 and -7 = -3 - 4. Knowing the importance of 4 in this sequence, our next step might be to turn every term into a multiple of 4 by adding 1:

$-1,3,7,\dots,83\xrightarrow{+1}0,4,8,\dots,84$

and since we're dealing with multiples of 4, a natural next step might be to divide every term by 4:

$0,4,8,\dots,84\xrightarrow{\div4}0,1,2,\dots,21$

And lastly, we can add 1 to every term to get our sequence into easily-countable form:

$0,1,2,\dots,21\xrightarrow{+1}1,2,3,\dots,22$

So, the sequence a has 22 terms.