The 10th term of an AP is - 37 and sum of its first 6 terms is - 27. Find the sum of its first eight terms.

Answer: - 63

Step-by-step explanation:

An arithmetic progression is such that the N-th element is equal to:

An = A1 + (n-1) * R

where A1 is the first term, and R is the difference between two consecutive terms.

We know that

A1 + 9*R = - 37

and

A1 + (A1 + R) + (A1 + 2R) + (A1 + 3R) + (A1 + 4R) + (A1 + 5R) = - 27

6*A1 + R (1 + 2 + 3 + 4 + 5) = - 27

6*A1 + 15*R = - 27

so we have two equations:

6*A1 + 15*R = - 27

A1 + 9*R = - 37

Let's find A1 and R.

First, isolate A1 in the second equation:

A1 = - 37 - 9*R

now replace it in the other equation and solve it for R.

6*A1 + 15*R = - 27

6 * (-37 - 9*R) + 15*R = - 27

-222 - 54*R + 15*R = - 27

-39*R = - 27 + 222 = 195

R = 195/-39 = - 5

Now we can find the value of A1:

A1 = - 37 - 9*R = - 37 - 9*-5 = 8

So now we want to calculate the sum of the first eight terms. we already know that the sum of the first six is - 27, so we need to add the seventh and the eigth.

A7 = 8 + 6*-5 = - 17

A8 = 8 + 7*-5 = - 22

Sum = - 27 - 17 - 22 = - 63