The 2nd term of an arithmetic progression is (1/2) and the sum of the first 14th terms is -70. Find (a) the common difference
(b) the last term.​

The common difference is -1.

The last term (the 14th term) is -11.5.

Step-by-step explanation:

In an arithmetic sequence, the second term is 0.5 and the sum of the first 14 terms is -70.

We want to determine the: (a) common difference and (b) the last term.

We can write an explicit formula to represent the sequence. An arithmetic sequence can be modeled by the formula:

Where a is the initial term, d is the common difference, and n represents the nth term.

Since the second term is 0.5:

Simplify:

The sum of an arithmetic sequence is given by the formula:

Where k is the number of terms and x_k is the last term.

Since the sum of the first 14 terms is -70, S = -70 and k = 14:

Using our explicit formula, the last term is:

Substitute:

Simplifiy:

Rewrite the equation for the second term:

Substitute:

Simplify:

Solve for d:

Hence, our common difference is -1.

Solve for a, the initial term:

So, our explicit formula is now:

So, the last term (which is 14) is:

answer:   x = 3 + √11     and     x = 3 - √11

step-by-step explanation:

x² - 6x + 9 = 11

(x - 3)² = 11

x - 3 = ±√11

    x = 3 ± √11

x = 3 + √11     and     x = 3 - √11

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