In an AP, the thirteenth term is 27, and the seventh term is three times the second term. Find the first term,?
a+12d = 27 , (#1)
a+6d = 3(a+d)
a+6d = 3a + 3d
3d = 2a
then 12d = 8a
in a+12d = 27
a + 8a = 27
9a = 27
a = 3
first term is 3
To find the first term of an arithmetic progression (AP), we need to use the given information about the 7th term and the 13th term.
Let's proceed step by step:
Step 1: Identify the formula for the nth term of an AP
The formula for finding the nth term of an AP is:
a + (n - 1)d
where 'a' is the first term and 'd' is the common difference.
Step 2: Use the given information to form equations
From the given information, we have:
a + 6d = 3(a + d) (since the 7th term is three times the second term)
a + 12d = 27 (since the 13th term is 27)
Step 3: Solve the equations simultaneously to find 'a' and 'd'
To solve the equations simultaneously, we can use the method of substitution or elimination. Let's use the substitution method here:
From equation 1, we have:
a = 3a + 6d - 3d
a = 3a + 3d
2a = 3d ...........(Equation 3)
Substitute equation 3 into equation 2:
3(2a) + 12d = 27
6a + 12d = 27
2a + 4d = 9 ...........(Equation 4)
Now, we have a system of two equations:
2a + 4d = 9
2a = 3d
Step 4: Solve the system of equations
Multiply equation 4 by 2:
4a + 8d = 18
Subtract equation 4 from equation 5:
(4a + 8d) - (4a + 4d) = 18 - 9
4d = 9
d = 9/4
Substitute the value of 'd' back into equation 3:
2a = 3(9/4)
2a = 27/4
a = 27/8
Therefore, the first term (a) of the arithmetic progression is 27/8.