From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!
Write the first four terms of the arithmetic sequence with a1 = 4 and d = 5.
To get from one term to the next we add d = 5. So
a2 = a1 + 5 = 9
a3 = a2 + 5 = 14
a4 = a3 + 5 = 19
We were given the first term, a1 = 4. The first four terms are
4, 9, 14, 19.
If a1 = 3 and d = 5, find a5.
To get from a1 to a5 takes four steps.
Each step has size d = 5.
If we start at a1 = 3 and take four steps of size d = 5, we land at
a5 = a1 + 4d
= 3 + 4(5)
If a1 = 12 and d = -2, find a11.
It takes 1 step to get to a2, 2 steps to get to a3, 3 steps to get to a4, etc. It must take 10 steps to get to a11.
Each step has size d = -2.
If we start at a1 = 12 and take 10 steps of size d = -2, we land at
a11 = a1 + 10d
= 12 + 10(-2)
An arithmetic sequence has a5 = 11 and a6 = 13. Find a1 and d.
If this is an arithmetic sequence, then d = 2 because that's the size of the step from a5 to a6. We must have taken 4 steps to get from a1 to a5. Working backwards by stepping down 2 each time, we can fill in a4, a3, a2, and finally a1. We find that a1 = 3.
Here's another way to do the example. Whatever a1 is, we took 4 steps of size 2 to get to a5.
a1 + 4(2) = a5.
Since we know a5 = 11, we can solve this equation for a1.
a1 + 4(2) = 11
a1 = 3
Thankfully, we got the same answer as before.
An arithmetic sequence has a10 = 52 and a11 = 56. Find a1 and d.
Since this is an arithmetic sequence, the step size d is
a11 – a10 = 4.
We could work backwards and find a9, a8, etc., but that's more steps than we really want to do. Instead, let's just think about working backwards. Whatever a1 is, we know it takes 9 steps of size 4 to get to a10. In symbols,
a1 + 9(4) = a10.
Since we know a10 = 52, we can solve this equation for a1:
a1 + 9(4) = 52
a1 = 52 – 36
a1 = 16
Find a1 and d for the arithmetic sequence with a2 = 8 and a4 = 28
To get from a2 to a4 takes two steps.
The difference between a4 and a2 is
a4 – a2 = 28 – 8 = 20.
In order for this difference of 20 to be split between two equally-sized steps, each step must have size 10.
This means d = 10.
Now we proceed as before. There's one step from a1 to a2. If d = 10 and a2 = 8, then
a1 + 1d = a2
a1 + (10) = 8
a1 = -2.
Find a1 and d for the arithmetic sequence with a50 = 14 and a100 = -136.
100 – 50 = 50
steps from a50 to a100. This means the difference
a100 – a50 = -136 – 14 = -150
must be divided into 50 separate steps. We find
Then we proceed as usual. We have 49 steps from a1 to a50, so
a1 + 49d = a50
a1 + 49(-3) = 14
a1 = 161.
To summarize, if we're given two terms am and an that aren't consecutive, we