The fourth term of an AP is 9 and the sum of the first ten terms is 60. Find the first term and the common difference.
Use your AP formulas ...
a+3d = 9
10/2 (2a+9d) = 60
Now solve as usual.
To find the first term and the common difference of an Arithmetic Progression (AP), we can use the formulas for the nth term and the sum of the first n terms of an AP.
Given information:
- The fourth term of the AP is 9: a + 3d = 9 (1), where a is the first term and d is the common difference.
- The sum of the first ten terms is 60: S10 = 60 (2).
Formula for the nth term of an AP:
an = a + (n-1)d
Formula for the sum of the first n terms of an AP:
Sn = (n/2)(2a + (n-1)d)
Let's solve the equations step by step:
Step 1: Finding the first term (a):
We can use equation (1) to express a in terms of d: a = 9 - 3d
Step 2: Finding the common difference (d):
Substitute the expression of a from step 1 into equation (2) and solve for d:
S10 = (10/2)(2(9 - 3d) + (10-1)d)
60 = 5(18 - 6d + 9d)
60 = 5(18 + 3d)
60 = 90 + 15d
15d = -30
d = -30/15
d = -2
Step 3: Finding the first term (a):
Substitute the value of d from step 2 into the expression of a from step 1:
a = 9 - 3(-2)
a = 9 + 6
a = 15
Therefore, the first term (a) of the AP is 15 and the common difference (d) is -2.