Math  /  Algebra

QuestionThe first three terms of a sequence are given. Write your answer as a decimal or whole number. Round to the nearest thousandth (if necessary). 255,250,245,255,250,245, \ldots
Find the 40 th term.

Studdy Solution

STEP 1

What is this asking? We're looking for the 40th number in a sequence of numbers that starts with 255, 250, and 245, and seems to be going down! Watch out! Don't forget that the first term is term number one, not term number zero.
It's easy to get tripped up on that!

STEP 2

1. Find the common difference.
2. Find an explicit formula.
3. Calculate the 40th term.

STEP 3

To find the common difference, we **subtract** any term from the term that comes after it.
Let's subtract the first term from the second term: 250255=5250 - 255 = -5.

STEP 4

So, our common difference is 5-5.
This means each term is **5 less** than the one before it!

STEP 5

The formula for an arithmetic sequence is: an=a1+(n1)da_n = a_1 + (n - 1) \cdot d, where ana_n is the nth term, a1a_1 is the **first term**, nn is the term number, and dd is the **common difference**.

STEP 6

We know that a1=255a_1 = 255 and d=5d = -5.
Let's **plug those values** into our formula: an=255+(n1)(5)a_n = 255 + (n - 1) \cdot (-5).

STEP 7

Let's **simplify** this formula a bit: an=2555n+5a_n = 255 - 5n + 5, which simplifies further to an=2605na_n = 260 - 5n.
Now we have a nice, clean formula!

STEP 8

We want to find the 40th term, so we're looking for a40a_{40}.
Let's **plug in** n=40n = 40 into our formula: a40=260540a_{40} = 260 - 5 \cdot 40.

STEP 9

Now, let's **calculate**: a40=260200=60a_{40} = 260 - 200 = 60.
So, the 40th term is **60**!

STEP 10

The 40th term of the sequence is 60.

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