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Problem 655

# If the 6 th term of an arithmetic progression is 8 and the $$11^{\text {th }}$$ term is $$-2$$, what is the 1 st term? What is the common difference?

Expert verified
The first term of the arithmetic progression is 18 and the common difference is -2.
See the step by step solution

## Step 1: Write the equations for the 6th and 11th terms

Using the formula of an arithmetic progression, we write the equations for the 6th term and the 11th term: 6th term: $$8 = a + (6-1)d$$ 11th term: $$-2 = a + (11-1)d$$

## Step 2: Solve for the common difference (d)

Let's use two variables for our ease: x = a + 5d (6th term equation) y = a + 10d (11th term equation) Subtract the first equation from the second equation to eliminate "a": (y - x) = (10d - 5d) -2 - 8 = 5d -10 = 5d d = -2

## Step 3: Solve for the first term "a"

Now, we can substitute the value of "d" into either equation to solve for "a": 8 = a + 5(-2) 8 = a - 10 a = 18 So, the first term of the arithmetic progression is 18, and the common difference is -2.

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