calculus question?

Question

find the first term and the common difference of the arithmetic sequence whose 10th term is -33 and 14th term is -53.

first term is:

common difference is:

Thank you for your help!

Answer 1

The nth term of an arithmetic progression is calculated as follows. Let a be the first term, d be the difference, and a[n] be the nth term of the sequence.

a[n] = a + (n - 1)d

We are given a[10] = -33, but by definition,
a[10] = a + (10 - 1)d
a[10] = a + 9d
So it follows a + 9d = -33

a[14] = -53, so
a[14] = a + (14 - 1)d
a[14] = a + 13d
So a + 13d = -53

Two equations, two unknowns.

a + 9d = -33
a + 13d = -53

Use elimination to solve.

-4d = 20
d = -5

Now that we know d = -5, we can get a.
a + 13d = -53
a + 13(-5) = -53
a - 65 = -53
a = 12

a = 12, d = -5
The first term is 12, the difference is -5.

To definitely prove these are the correct values, here's the partial sequence:

12, 7, 2, -3, -8, -13, -18, -23, -28, -33, -38, -43, -48, -53

Answer 2

a+9d=-33
a+13d=-53

d=-5
a+9(-5)=-33

a=12

Answer 3

what in the world???!!! man i just dont get mad...its soooo hard!!!