The sum of the 3rd and 7th terms of an a.p is 36, and the 9th term is 37. Find the a.p
Just translate into English:
"sum of the 3rd and 7th terms of an a.p is 36"
---> a+2d + a+6d = 36
2a + 8d = 36
a + 4d = 18 ***
"he 9th term is 37" ---> a+8d = 37
a = 37-8d
plug that into *** and solve.
Use your definitions to state the AP
don't really get the answer
please explain it clearly God bless you
To find the arithmetic progression (a.p.), we need to find the common difference (d) and the first term (a1).
Let's start by identifying the given information:
1. The sum of the 3rd and 7th terms is 36.
2. The 9th term is 37.
First, we need to find the common difference (d).
The formula for finding the sum of two terms in an arithmetic progression is:
Sum = (n/2) * [2a + (n-1)d]
Using this formula, we can find the sum of the 3rd and 7th terms:
36 = (2/2) * [2a + (2-1)d] = 2a + d ----(1)
We also know that the 9th term is 37. Using the formula for the nth term in an a.p., we have:
a9 = a1 + (n-1)d
37 = a1 + (9-1)d
37 = a1 + 8d ----(2)
Now we have a system of equations, (1) and (2), with two variables (a and d). We can solve the system to find the values of a and d.
Subtracting equation (1) from equation (2), we get:
37 - 36 = a1 + 8d - (2a + d)
1 = -a + 7d
a - 7d = -1 ----(3)
Now, let's solve equation (1) and equation (3) simultaneously.
Multiplying both sides of equation (1) by 7, we get:
7(2a + d) = 7 * 36
14a + 7d = 252 ----(4)
Now we have equations (3) and (4) as a system of linear equations. We can solve this system by substitution or elimination.
Multiplying equation (3) by 14, we get:
14a - 98d = -14 ----(5)
Adding equation (4) and equation (5), we have:
14a + 7d + 14a - 98d = 252 - 14
28a - 91d = 238 ----(6)
Now we can solve equation (6) for 'a'.
From equation (3), we know that a - 7d = -1. Rearranging it, we have a = 7d - 1. Substituting this into equation (6), we get:
28(7d - 1) - 91d = 238
196d - 28 - 91d = 238
105d = 266
d = 266/105
d = 2.54 (rounded to two decimal places)
Now that we have the value of 'd', we can substitute it back into equation (3) to find 'a'.
a - 7(2.54) = -1
a - 17.78 = -1
a = 16.78 (rounded to two decimal places)
Therefore, the first term (a1) of the arithmetic progression is approximately 16.78 and the common difference (d) is approximately 2.54.
Hence, the arithmetic progression is: 16.78, 19.32, 21.86, 24.4, 26.94, ...