The sum of first m terms of an A.P is 5m²+3m. If its nth term is 168. Find the value of m. Also find the 20th term.
m/2 (2a+(m-1)d) = 5m^2+3m
Since I have no idea what n is supposed to be, I'll guess you meant m. So, that means that
a+(m-1)d = 168
m/2 (a+168) = 5m^2+3m
a+168 = 10m+6
a+162 = 10m
So, let's take a=8, making m=17
8+16d=168
d=10
5m^2+3m = 17(5*17+3) = 1496
17/2 (16+16*10) = 1496
So, one sequence is
8, 18, 28, 38, ...
T20 = 8+19*10 = 198
But, you can see that there are other such sequences as well.
To find the value of 'm' and the 20th term of the arithmetic progression (A.P.), we will use the given information and the formulas involved.
First, let's find the value of 'm':
We are given the sum of the first 'm' terms of the A.P., which is 5m² + 3m. The formula for the sum of the first 'n' terms of an A.P. is:
Sn = (n/2) * [2a + (n-1)d]
Where Sn represents the sum of the first 'n' terms, 'a' is the first term, 'd' is the common difference, and 'n' is the number of terms.
Comparing this with the given sum (5m² + 3m), we can equate the two expressions:
5m² + 3m = (m/2) * [2a + (m-1)d]
Since the A.P. does not have any specific values for 'a' and 'd', we need to analyze the other given information.
We know that the nth term of the A.P. is 168. The formula to find the nth term is:
an = a + (n - 1) * d
Comparing this with the given value (168), we can equate the two expressions:
168 = a + (n - 1) * d
We have two equations:
1) 5m² + 3m = (m/2) * [2a + (m-1)d]
2) 168 = a + (n - 1) * d
Let's solve these equations simultaneously to find the values of 'm' and 'a'.
From the second equation, we can express 'a' in terms of 'd':
a = 168 - (n - 1) * d
Now, substitute this value of 'a' into the first equation:
5m² + 3m = (m/2) * [2 * (168 - (n - 1) * d) + (m - 1) * d]
Simplify this equation and solve for 'm'.
Once you find the value of 'm', substitute it into the expression for the n-th term to find the 20th term.