The sum of the first 10 terms of an arithmetic sequence is 530. What is the first term if the last term is 80? What is the common difference?

An arithmetic sequence is a sequence of numbers such that the difference of any two succeeding terms of the sequence is a constant while an arithmetic series is the sum of an arithmetic sequence. We find the sum by adding the first, a and last term, an, divide by 2 in order to get the mean of the two values and then multiply by the number of values.

Step by Step Solution:

Write the given.

S = 530 (sum of the first 10 terms of the arithmetic sequence)

a = 80 (10th term of the arithmetic sequence)

Identify what is asked.

a = ____ (first term of the arithmetic sequence)

d = ____ (common difference of the arithmetic sequence)

Identify the formula to be used.

a = a + (n – 1)d

S = [2a + (n – 1)d]n/2

Write the solution.

a = a + (10 – 1)d

80 = a + 9d

Rename the equation to a by adding -9d to both sides of the equation. Thus, 80 + (-9d) = a + 9d + (-9d) which when simplified is 80 – 9d = a

Substitute the derived equation for a in the equation for the sum of the arithmetic sequence to solve for the common difference.

S = [2a + (n – 1)d]n/2

530 = [2(80 – 9d) + (10 – 1)d]10/2

530 = [160 – 18d + 9d]5

530 = [160 – 9d]5

530 = 800 – 45d

Combine the similar terms.

530 + (-800) = – 45d

Divide both sides by -45 to solve for d.

6 = d . Therefore, the common difference is 6.

Now, find the first term of the arithmetic sequence using the formula: 80 – 9d = a

80 – 9(6) = a

80 – 54 = a

26 = a . Therefore, the first term of the arithmetic sequence is 26.

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