What is the 32nd term of an arithmetic sequence where the first term is 32 and the ninth term is 120?

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Finding the 32nd Term of an Arithmetic Sequence

To determine the 32nd term of the arithmetic sequence where the first term is 32 (denoted as a1 = 32) and the ninth term is 120 (denoted as a9 = 120), we can follow these steps:

Step 1: Understand the Formula

The general formula for the nth term of an arithmetic sequence is given by:

an = a1 + (n - 1)d

Here, d represents the common difference between consecutive terms, and n is the term number.

Step 2: Find the Common Difference

We know:

  • a1 = 32
  • a9 = 120

Using the formula for the ninth term:

a9 = a1 + (9 - 1)d = 32 + 8d

We set it equal to 120:

32 + 8d = 120

Now, we solve for d:

8d = 120 - 32 

8d = 88 

d = 88 / 8 

d = 11

Step 3: Calculate the 32nd Term

Now that we know the common difference d = 11, we can find the 32nd term:

a32 = a1 + (32 - 1)d 

= 32 + 31 * 11 

= 32 + 341 

= 373

Conclusion

Therefore, the 32nd term of the arithmetic sequence is 373.

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