In an A.P , the sum of first ten terms is (-80) and the sum of its next ten terms is (-280)Find the A.P
To find the arithmetic progression (A.P) given the sum of its first ten terms and the sum of its next ten terms, we can use the formula for the sum of an A.P.
The formula for the sum of an A.P is:
Sn = (n/2)(2a + (n-1)d)
Where:
Sn = Sum of n terms
n = Number of terms
a = First term
d = Common difference
Let's solve this in two steps:
Step 1: Use the given information to set up two equations.
Given that the sum of the first ten terms is -80, we can write:
S10 = (-80) = (10/2)(2a + (10-1)d)
Similarly, the sum of the next ten terms is -280, so we have:
S20 = (-280) = (20/2)(2a + (20-1)d)
Step 2: Solve the equations simultaneously to find the values of a and d.
By dividing the equation for S20 by the equation for S10, we eliminate the common term (2a + 9d) and solve for d:
(-280)/(-80) = (20/2)/(10/2)
Simplifying, we get:
7/2 = 2
Now, substitute the value of d back into the equation for S10 to solve for a:
(-80) = (10/2)(2a + 9(2))
Simplifying further:
(-80) = 5(2a + 18)
(-80) = 10a + 90
10a = -170
a = -17
Therefore, the first term of the A.P (a) is -17, and the common difference (d) is 2. Thus, the A.P is -17, -15, -13, -11, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21.
So plug in your formula. If S1 is the sum of the 1st ten terms, and S2 is the sum of the 1st twenty terms, then you have
S1 = 10/2 (2a+9d) = -80
S2 = 20/2 (2a+19d) - S1 = -280
Rearranged a bit, that's
10a + 45d = -80
10a + 145d = -280
Looks like
a = 1
d = -2