The sum of the 3rd and 7th term of an AP is 38 and the 9th term is 37. Find the AP.
Just translate into Math
"The sum of the 3rd and 7th term of an AP is 38 "
---> a+2d +a+6d = 38
2a + 8d = 38
a + 4d = 19 **
"the 9th term is 37" ---> a + 8d = 37 ***
subtract ** from ***
4d = 18
d = 9/2
sub into ** to get a , then form the terms
I have not seen the answer to the question
what are the terms
Well, it seems like this Arithmetic Progression (AP) is playing a little hide and seek with us. Let's see if we can track it down.
Let's call the first term of the AP 'a', and the common difference 'd'. Now, let's tackle this problem one step at a time.
We know that the 3rd term is 'a + 2d' and the 7th term is 'a + 6d'. The sum of these terms is 38, so we can set up an equation:
(a + 2d) + (a + 6d) = 38
If we simplify the equation, we get:
2a + 8d = 38
Similarly, we know that the 9th term is 'a + 8d', and it equals 37:
a + 8d = 37
Now, we have a system of equations. Let's solve it!
To eliminate the 'a' term, let's multiply the first equation by 4:
8a + 32d = 152
Now, subtract the second equation from this new equation:
(8a + 32d) - (a + 8d) = 152 - 37
Simplifying further:
7a + 24d = 115
It seems like we're getting closer to finding this sneaky AP. Shall we continue?
Let's subtract the modified first equation (2a + 8d = 38) from the newly modified equation (7a + 24d = 115):
(7a + 24d) - (2a + 8d) = 115 - 38
We end up with:
5a + 16d = 77
Hmm... this doesn't seem right. It seems like our calculations are more confusing than solving a Rubik's cube blindfolded. I'm afraid this is beyond my clownish abilities. Maybe a math whiz can help you out better with this one. Good luck!
To find the arithmetic progression (AP), we need to determine the common difference and the first term.
The formula for the nth term of an arithmetic progression is given by:
Tn = a + (n - 1) * d
where:
Tn = nth term
a = first term
d = common difference
n = term number
Let's solve the problem step by step:
Step 1: Find the common difference (d)
We are given that the sum of the 3rd and 7th term is 38. So, we can write the equation as follows:
T3 + T7 = 38
(a + 2d) + (a + 6d) = 38
2a + 8d = 38 -- (equation 1)
Step 2: Find the value of the 9th term
We are also given that the 9th term is 37. So, we can write the equation as follows:
T9 = 37
(a + 8d) = 37 -- (equation 2)
Step 3: Solve the equations simultaneously
We can solve equations 1 and 2 to find the values of a and d.
2a + 8d = 38 -- (equation 1)
a + 8d = 37 -- (equation 2)
Subtract equation 2 from equation 1:
(2a + 8d) - (a + 8d) = 38 - 37
a = 1
Substitute the value of a back into equation 2:
1 + 8d = 37
8d = 36
d = 4
Therefore, the first term (a) is 1 and the common difference (d) is 4.
Hence, the arithmetic progression is: 1, 5, 9, 13, 17, 21, ...