The first three terms of an A.P. are p-5,p-3 and 2p-15.If the sum of the first n terms of the progression is 180,find the value of n.
Answer:n=10
well, now you know that
d = p-3 - (p-5) = 2
2p-15 - (p-3) = 2, so p=14
so, a = p-5 = 9
n/2 (2*9 + (n-1)*2) = 180
n = 10
Ohh..thank u ^^
To find the value of n in an arithmetic progression (A.P.), we need to use the given information about the first three terms and the sum of the terms.
Given:
The first term, a = p - 5
The second term, b = p - 3
The third term, c = 2p - 15
The sum of the first n terms, Sₙ = 180
In an arithmetic progression, the common difference (d) between consecutive terms is the same. We can use this information to find the value of d.
Step 1: Find the common difference (d):
The second term can be found by adding the common difference (d) to the first term:
b = a + d
p - 3 = (p - 5) + d
p - 3 = p - 5 + d
2 = d
So, the common difference (d) is 2.
Step 2: Find the value of p:
Now, substitute the value of d (2) into the third term equation:
c = b + d
2p - 15 = (p - 3) + 2
2p - 15 = p - 1
p = 14
So, the value of p is 14.
Step 3: Find the value of n:
Since Sₙ represents the sum of the first n terms, we can use the formula for the sum of an arithmetic progression:
Sₙ = (n/2) * (2a + (n - 1)d)
Substitute the values of a, d, and Sₙ into the equation to solve for n:
180 = (n/2) * (2(p-5) + (n - 1)(2))
180 = (n/2) * (2(p-5) + 2(n-1))
180 = (n/2) * (2p - 10 + 2n - 2)
Simplify the equation:
180 = (n/2) * (2p + 2n - 12)
180 = n(p + n - 6)
Substitute the value of p (14) into the equation:
180 = n(14 + n - 6)
180 = n(8 + n)
180 = 8n + n²
Rearrange the equation and solve for n:
n² + 8n - 180 = 0
Factorize the quadratic equation:
(n + 18)(n - 10) = 0
Set each factor equal to zero to find the possible values of n:
n + 18 = 0 or n - 10 = 0
n = -18 or n = 10
Since the value of n cannot be negative in this context, the value of n is 10.
Therefore, n = 10 is the correct answer.