The sum of the 3rd and 7th term of an A.P is 38 and the 9th term is 37..Find the first term and common difference
just set up the equations:
a+2d + a+6d = 38
a+8d = 37
Now just solve for a and d.
To find the first term and common difference of an arithmetic progression (A.P), we can use the formula for the nth term of an A.P, which is given by:
an = a + (n - 1)d,
where an is the nth term, a is the first term, n is the term number, and d is the common difference.
Let's solve the given problem step by step:
Step 1: Finding the values for the 3rd and 7th terms:
Given that the sum of the 3rd and 7th terms is 38, we can write the following equation:
a3 + a7 = 38.
Using the formula for the nth term, we can rewrite this equation as:
a + 2d + a + 6d = 38.
Simplifying further, we get:
2a + 8d = 38.
Step 2: Finding the value for the 9th term:
Given that the 9th term is 37, we can write the following equation:
a9 = 37.
Using the formula for the nth term, we can rewrite this equation as:
a + 8d = 37.
Step 3: Solving the system of equations:
We now have two equations with two variables:
Equation 1: 2a + 8d = 38,
Equation 2: a + 8d = 37.
We can solve this system of equations by substitution or elimination method.
Let's use the elimination method:
Subtracting Equation 2 from Equation 1, we get:
(2a + 8d) - (a + 8d) = 38 - 37,
a = 1.
Step 4: Substitute the value of a back into one of the equations to find d:
Using Equation 2:
a + 8d = 37,
1 + 8d = 37,
8d = 37 - 1,
8d = 36,
d = 36/8,
d = 4.5.
Therefore, the first term (a) is 1 and the common difference (d) is 4.5.