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Problem 660

# Given that the first term of an arithmetic sequence is 56 and the seventeenth term is 32 , find the tenth term and the twenty - fifth term.

Expert verified
The tenth term of the arithmetic sequence is $$\dfrac{85}{2}$$, and the twenty-fifth term is 20.
See the step by step solution

## Step 1: Define the arithmetic sequence formula

The formula for the nth term of an arithmetic sequence is given by: $$a_n = a_1 + (n - 1)d$$ where $$a_n$$ is the nth term, $$a_1$$ is the first term, $$n$$ is the term number, and $$d$$ is the common difference between consecutive terms.

## Step 2: Calculate the common difference

We are given the first term ($$a_1 = 56$$) and the seventeenth term ($$a_{17} = 32$$). We can use the arithmetic sequence formula to find the common difference (d): $$a_{17} = a_1 + (17 - 1)d$$ Substitute the given values into the formula: $$32 = 56 + (16)d$$ Now, we solve for d: $$16d = 32 - 56$$ $$16d = -24$$ $$d = -\dfrac{24}{16} = -\dfrac{3}{2}$$ The common difference, d, is $$-\dfrac{3}{2}$$

## Step 3: Find the tenth term

Now that we have the common difference, we can find the tenth term, $$a_{10}$$, using the arithmetic sequence formula: $$a_{10} = a_1 + (10 - 1)d$$ Substitute the values: $$a_{10} = 56 + (9)(-\dfrac{3}{2})$$ $$a_{10} = 56 - \dfrac{27}{2}$$ $$a_{10} = \dfrac{85}{2}$$ The tenth term, $$a_{10}$$, is $$\dfrac{85}{2}$$.

## Step 4: Find the twenty-fifth term

Similarly, we can find the twenty-fifth term, $$a_{25}$$, using the arithmetic sequence formula: $$a_{25} = a_1 + (25 - 1)d$$ Substitute the values: $$a_{25} = 56 + (24)(-\dfrac{3}{2})$$ $$a_{25} = 56 - 36$$ $$a_{25} = 20$$ The twenty-fifth term, $$a_{25}$$, is 20. To summarize, the tenth term of the arithmetic sequence is $$\dfrac{85}{2}$$, and the twenty-fifth term is 20.

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