The 10th term of an A.P is -27 and the 5th term is -12.what is 18th term?also find sum of it's 25terms
To find the 18th term of an arithmetic progression (A.P), we need to find the common difference (d) first. We can find the common difference using the given 10th and 5th terms.
Given:
5th term (a5) = -12
10th term (a10) = -27
The formula to find the nth term (an) is:
an = a1 + (n - 1) * d
Using this formula, we can find the common difference:
-27 = a1 + (10 - 1) * d (Substituting a10 as -27)
-12 = a1 + (5 - 1) * d (Substituting a5 as -12)
Simplifying the equations, we get:
a1 + 9d = -27 (equation 1)
a1 + 4d = -12 (equation 2)
By subtracting equation 2 from equation 1, we can eliminate a1 and solve for d:
(a1 + 9d) - (a1 + 4d) = -27 - (-12)
5d = -15
d = -15/5
d = -3
Now that we have the common difference (d = -3), we can find the 18th term (a18) using the formula for the nth term:
a18 = a1 + (18 - 1) * d
= a1 + 17d
To find a1, we can use the 5th term (a5):
a1 = a5 - (5 - 1) * d
= -12 - 4 * (-3)
= -12 + 12
= 0
Now we can find the 18th term:
a18 = 0 + 17 * (-3)
= 0 - 51
= -51
Therefore, the 18th term of the A.P is -51.
To find the sum of the first 25 terms of the A.P (S25), we can use the formula:
S25 = (n/2)(2a1 + (n-1)d)
Substituting the known values:
S25 = (25/2)(2(0) + (25-1)(-3))
= (25/2)(0 + 24 * -3)
= (25/2)(0 - 72)
= (25/2)(-72)
= (25 * -72) / 2
= -1800
Therefore, the sum of the first 25 terms of the A.P is -1800.
To find the 10th term of an arithmetic progression (A.P.), we need to find the common difference (d) between consecutive terms. The formula to find the nth term of an A.P. is:
𝑎𝑛 = 𝑎 + (𝑛−1)𝑑
where 𝑎 is the first term, 𝑛 is the position of the term, and 𝑎𝑛 is the nth term.
Given that the 5th term is -12, we have:
𝑎5 = 𝑎 + (5−1)𝑑
-12 = 𝑎 + 4𝑑 ... (equation 1)
Given that the 10th term is -27, we have:
𝑎10 = 𝑎 + (10−1)𝑑
-27 = 𝑎 + 9𝑑 ... (equation 2)
Now we have a system of two equations with two variables, 𝑎 and 𝑑. We can solve this system to find the values of 𝑎 and 𝑑. Subtracting equation 1 from equation 2, we have:
-27 - (-12) = 𝑎 + 9𝑑 - 𝑎 - 4𝑑
-15 = 5𝑑
𝑑 = -15/5
𝑑 = -3
Substituting the value of 𝑑 into equation 1, we can find 𝑎:
-12 = 𝑎 + 4(-3)
-12 = 𝑎 - 12
𝑎 = 0
Therefore, the first term (𝑎) is 0 and the common difference (𝑑) is -3.
To find the 18th term (𝑎18), we can use the formula:
𝑎18 = 𝑎 + (18−1)𝑑
𝑎18 = 0 + 17(-3)
𝑎18 = -51
So, the 18th term is -51.
To find the sum of the first 25 terms, we can use the formula for the sum of an arithmetic series:
𝑆𝑛 = (𝑛/2)(𝑎 + 𝑎𝑛)
where 𝑆𝑛 is the sum of the first 𝑛 terms.
Substituting the values into the sum formula:
𝑆25 = (25/2)(0 + (-51))
𝑆25 = (25/2)(-51)
𝑆25 = -637.5
Therefore, the sum of the first 25 terms is -637.5.