THE 10TERM OF AN AP IS -27 AND THE 5TH TERM IS -12 WHAT IS THE 18th term find also the sum of its 25TERM
To find the 18th term of an arithmetic progression (AP) and the sum of its 25th term, we need to use the formula for the nth term of an AP and the formula for the sum of the first n terms of an AP.
Let's start with finding the common difference (d) of the AP using the given information. We are given that the 10th term (a₁₀) is -27 and the 5th term (a₅) is -12.
We can use the formula to find the nth term of an AP:
aₙ = a₁ + (n-1)d
Plugging in the values, we get:
-27 = a₁₀ = a₁ + 9d ...........(1)
-12 = a₅ = a₁ + 4d ............(2)
Now, let's solve these two equations to find the value of a₁ and d.
Subtract equation (2) from equation (1):
-27 - (-12) = a₁ + 9d - a₁ - 4d
-27 + 12 = 5d
-15 = 5d
d = -15/5
d = -3
Substitute the value of d into equation (2) to find a₁:
-12 = a₁ + 4(-3)
-12 + 12 = a₁
a₁ = 0
Now, we have a₁ = 0 and d = -3. We can use these values to find the 18th term and the sum of the 25 terms.
To find the 18th term (a₁₈) of the AP, we can use the formula:
aₙ = a₁ + (n-1)d
Substituting the values, we get:
a₁₈ = 0 + (18-1)(-3)
a₁₈ = -3(17)
a₁₈ = -51
So, the 18th term is -51.
To find the sum of the first 25 terms (S₂₅) of the AP, we can use the formula for the sum of an AP:
Sₙ = (n/2)(2a₁ + (n-1)d)
Substituting the values, we get:
S₂₅ = (25/2)(2(0) + (25-1)(-3))
S₂₅ = (25/2)(0 + 24(-3))
S₂₅ = (25/2)(0 - 72)
S₂₅ = (25/2)(-72)
S₂₅ = (25)(-36)
S₂₅ = -900
So, the sum of the first 25 terms is -900.
a+9d = -27
a+4d = -12
Solve for a and d, then find
T18 = a+17d
S25 = 25/2 (2a + 24d)