The 10th and 10th terms of an Ap is -27 and -12, what is the 18th term, find the sum of its 25 terms
The 10th and 10th terms ... ???
after you fix your typo and solve for a and d, then
a_18 = a+17d
S25 = 25/2 (2a + 24d)
To find the 18th term of an arithmetic progression (AP), we need to know the first term (a) and the common difference (d).
Given that the 10th term (a₁₀) is -27 and the 10th term (a₁₀) is -12, we can use these two terms to find the common difference.
The formula to find the nth term of an AP is:
aₙ = a + (n - 1) * d
Using the values we have:
-12 = a + (10 - 1) * d
-27 = a + (10 - 1) * d
Simplifying these equations, we can subtract them from each other to eliminate 'a' and solve for 'd':
-12 - (-27) = (10 - 1) * d
15 = 9d
d = 15/9
d = 5/3
Now that we have the common difference (d), we can find the first term (a) using the 10th term (a₁₀):
-27 = a + (10 - 1) * (5/3)
-27 = a + 9 * (5/3)
-27 = a + 45/3
-27 = a + 15
a = -42
Now, we can find the 18th term (a₁₈) using the formula:
a₁₈ = a + (18 - 1) * d
a₁₈ = -42 + (18 - 1) * (5/3)
a₁₈ = -42 + 17 * (5/3)
a₁₈ = -42 + 85/3
a₁₈ = -42 + 28 1/3
a₁₈ = -42 + 85/3
a₁₈ = -42 + 255/9
a₁₈ = -42 + 85/9
a₁₈ = -378/9 + 85/9
a₁₈ = -293/9
Therefore, the 18th term of the arithmetic progression is -293/9.
To find the sum of the first 25 terms, we can use the formula for the sum of an AP:
Sₙ = (n/2) * (2a + (n - 1) * d)
Plugging in the values, we get:
S₂₅ = (25/2) * (2(-42) + (25 - 1) * (5/3))
S₂₅ = (25/2) * (-84 + 24 * (5/3))
S₂₅ = (25/2) * (-84 + 40 * (5/3))
S₂₅ = (25/2) * (-84 + 200/3)
S₂₅ = (25/2) * (-504/6 + 200/3)
S₂₅ = (25/2) * ((-504 + 400)/6)
S₂₅ = (25/2) * (-104/6)
S₂₅ = (25/2) * (-52/3)
S₂₅ = 25 * (-26/3)
S₂₅ = -650/3
Therefore, the sum of the first 25 terms of the arithmetic progression is -650/3.