The first term of an AP is -5 and the fifteenth term is double the third term. Find the eleventh term of the progression.
So a= -5 and
a+14d = 2(a+2d)
you know a, so find d then evaluate
a + 10d
To find the eleventh term of the arithmetic progression (AP), we need to determine the common difference and then calculate the term using the formula for the nth term of an AP.
Step 1: Find the common difference (d):
We are given that the first term (a₁) is -5 and the fifteenth term (a₁₅) is double the third term (a₃).
a₁₅ = 2 * a₃
Using the formula for the nth term of an AP:
a₁₅ = a₁ + (15 - 1) * d
a₃ = a₁ + (3 - 1) * d
Since a₁₅ = 2 * a₃, we can write:
a₁ + (15 - 1) * d = 2 * (a₁ + (3 - 1) * d)
-5 + 14d = 2 * (-5 + 2d)
Simplifying the equation:
-5 + 14d = -10 + 4d
14d - 4d = -10 + 5
10d = -5
d = -5/10
d = -1/2
So, the common difference (d) is -1/2.
Step 2: Calculate the eleventh term (a₁₁):
We can now use the formula for the nth term of an AP to find the eleventh term (a₁₁):
a₁₁ = a₁ + (11 - 1) * d
a₁₁ = -5 + 10 * (-1/2)
Simplifying the equation:
a₁₁ = -5 - 5
a₁₁ = -10
Therefore, the eleventh term of the arithmetic progression is -10.