the 3rd term of an A.p is 9 while the 11th term is -7. Solve for the 1st five terms of the A.p
Solution for the first 5 terms
a+2d = 9
a+10d = -7
so, d = -2 and a=13
so now just write out the terms
is to knowledge
The solving
Solve
What is the answer
To solve for the first five terms of an arithmetic progression (A.P.), we need to determine the common difference (d) and the first term (a1).
Given that the 3rd term (a3) is 9 and the 11th term (a11) is -7, we can use the formula for the nth term of an A.P. to calculate the common difference.
The formula for the nth term of an A.P. is:
an = a1 + (n - 1) * d
Substituting the values for a3 and a11 into the formula, we can form two equations as follows:
a3 = a1 + (3 - 1) * d ...(1)
a11 = a1 + (11 - 1) * d ...(2)
From equation (1), we have:
9 = a1 + 2d ...(3)
From equation (2), we have:
-7 = a1 + 10d ...(4)
Now we have a system of two equations with two variables (a1 and d). We can solve this system to find the values.
First, let's multiply equation (3) by 5 and equation (4) by 1 to eliminate a1. This gives us:
5 * (9) = 5a1 + 10d ...(5)
1 * (-7) = a1 + 10d ...(6)
Expanding and rearranging equation (5), we get:
45 = 5a1 + 10d ...(7)
Equation (6) remains the same:
-7 = a1 + 10d ...(6)
Now we can subtract equation (6) from equation (7) to eliminate a1:
(45) - (-7) = (5a1 + 10d) - (a1 + 10d)
52 = 4a1
Dividing both sides by 4, we have:
a1 = 52/4
a1 = 13
Now we know the value of a1, which is 13. We can substitute this back into equation (3) to solve for d:
9 = 13 + 2d
2d = 9 - 13
2d = -4
d = -4/2
d = -2
We have found the values of the first term (a1 = 13) and the common difference (d = -2).
To find the first five terms of the A.P., we can use the formula for the nth term:
an = a1 + (n - 1) * d
Using this formula, we can calculate the first five terms as follows:
a1 = 13
a2 = a1 + (2 - 1) * d = 13 + (2 - 1) * (-2) = 13 - 2 = 11
a3 = a1 + (3 - 1) * d = 13 + (3 - 1) * (-2) = 13 - 4 = 9
a4 = a1 + (4 - 1) * d = 13 + (4 - 1) * (-2) = 13 - 6 = 7
a5 = a1 + (5 - 1) * d = 13 + (5 - 1) * (-2) = 13 - 8 = 5
Therefore, the first five terms of the arithmetic progression are 13, 11, 9, 7, and 5.