The sixth team of an arithmetic progression is -3 and the sum of the first ten terms is -10 find the first term and the common difference.
To find the first term and the common difference of an arithmetic progression, we can use the formulas:
nth term = first term + (n - 1) * common difference
sum of n terms = (n/2) * (2 * first term + (n - 1) * common difference)
Given that the sixth term is -3, we can substitute this value into the formula for the nth term:
-3 = first term + (6 - 1) * common difference
-3 = first term + 5 * common difference
Now, let's use the second piece of information given, which is that the sum of the first ten terms is -10. We'll substitute this into the formula for the sum of n terms:
-10 = (10/2) * (2 * first term + (10 - 1) * common difference)
-10 = 5 * (2 * first term + 9 * common difference)
-10 = 10 * first term + 45 * common difference
So, we have two equations:
-3 = first term + 5 * common difference (1)
-10 = 10 * first term + 45 * common difference (2)
Now, we can solve these equations simultaneously to find the values of the first term and the common difference.
First, let's solve equation (1) for the first term in terms of the common difference:
first term = -3 - 5 * common difference
Now, substitute this expression for the first term in equation (2), and solve for the common difference:
-10 = 10 * (-3 - 5 * common difference) + 45 * common difference
-10 = -30 - 50 * common difference + 45 * common difference
-10 = -30 - 5 * common difference
-10 + 30 = -5 * common difference
20 = 5 * common difference
common difference = 20 / 5
common difference = 4
Now, substitute the value of the common difference back into equation (1) and solve for the first term:
-3 = first term + 5 * 4
-3 = first term + 20
first term = -3 - 20
first term = -23
Therefore, the first term of the arithmetic progression is -23 and the common difference is 4.