Given two terms of the arithmetic​ sequence, a9 = 30 and a14 = -15​, find a1 and d.

Answer 1

An arithmetic sequence means you are adding some number to each term to get to the next term. Based on the information in this problem, we can sketch the following:

Can you see that to go from a9 to a14, we need to add "d" five times therefore:

30 + 5d = -15

5d = -15 - 30

5d = -45

d = -45/5

d = -9

Using the equation for an arithmetic sequence: a_n = a₁ + d(n - 1) we can use the fact that a₉ = 30 and plug the values of n = 9 and a₉ = 30 into the equation:

30 = a₁ + (-9)(9 - 1)

30 = a₁ + (-9)(8)

30 = a₁ + -72

30 + 72 = a₁

a₁ = 102

Answer 2

There are two ways to figure this out - one is a more intuitive way, based on the understanding of what arithmetic sequences are, and one is by using a system of equations (this second method is a lot quicker!).

Method 1: "Intuitive" way

List out all of the numbers and place 30 in the 9th position and -15 in the 14th position.

_____, _____, _____, _____, _____, _____, _____, _____, 30, _____, _____, _____, _____, -15

As you can see, the 14th number is smaller than the 9th number... so that tells you that the common difference is negative.

The difference from -15 to 30 is -45 and it took 5 numbers to get there... that tells you that the common difference is -45/5 = -9.

Since we know that our 9th number is 30, if we subtract -9 eight times (same as adding 9 eight times) from 30, we will get our first term. That gives us 102.

So a₁ = 102, d = -9

If you thought that method seems a bit unwieldly and awkward... especially as numbers get bigger, you're right! Here's a faster method:

Method 2: Using a system of equations

In this case, it helps to remember that the general formula for the nth term of an arithmetic sequence can be given by: a_n = a₁ + (n - 1) d

Since we know that the 9th term is 30 and the 14th term is -15, we can substitute the values as follows:

30 = a₁ + (9 - 1) d. --> 30 = a₁ + 8d [Equation 1]

-15 = a₁ + (14 - 1)d -->. -15 = a₁ + 13d [Equation 2]

By subtracting the second equation from the first equation [Eq 1 - Eq 2], we can eliminate the a₁ term and get:

45 = -5d. ---> d = -9

Now, we can substitute the d back into one of the equations (either Equation 1 OR Equation 2) to solve for a_1:

(I'll substitute into Equation 1). 30 = a₁ + 8(-9). --> 30 = a₁ - 72 --> 102 = a₁

So, once again: a₁ = 102, d = -9

Hope that's helpful! Feel free to reach out to schedule regular lesson for more support!