The 5th term of an A.P is 3times the first.find the sum of the first 8 term of progression.given that 1/4 of 5th termis9
3 a = a + d(4) or 2a = 4d or a = 2d
fifth term is
a + 4 d
which is 3 a of course
(1/4)(3a) = 9
3 a = 4*9
a = 12
so d = 6
12, 18 , 24 ....
you can find the sum of first 8 I am sure
I have not seen the answer to my question
To find the sum of the first 8 terms of an arithmetic progression (A.P.), we need to know the first term (a), the common difference (d), and the number of terms (n).
Let's solve this step by step:
Step 1: Identify the given information.
- The 5th term of the A.P. is 3 times the first term.
- 1/4 of the 5th term is 9.
Step 2: Use the given information to determine the first term (a) and the common difference (d).
Let the first term be 'a' and the common difference be 'd'.
From the given information, we can write two equations.
Equation 1: 5th term = a + 4d = 3a (since the 5th term is 3 times the first term)
Equation 2: (1/4) * (a + 4d) = 9 (since 1/4 of the 5th term is 9)
Simplifying Equation 2:
(1/4) * (a + 4d) = 9
(a + 4d) = 9 * 4
a + 4d = 36 (Equation 3)
Substituting Equation 3 into Equation 1:
36 = 3a
a = 36 / 3
a = 12 (first term)
Substituting a = 12 into Equation 3:
12 + 4d = 36
4d = 36 - 12
4d = 24
d = 24 / 4
d = 6 (common difference)
Therefore, the first term (a) is 12, and the common difference (d) is 6.
Step 3: Calculate the sum of the first 8 terms using the formula for the sum of an arithmetic progression.
The sum of the first n terms (Sn) of an A.P. can be calculated using the formula:
Sn = (n/2) * (2a + (n-1)d)
For the first 8 terms (n = 8), we can substitute the values into the formula:
S8 = (8/2) * (2(12) + (8-1)(6))
= 4 * (24 + 7 * 6)
= 4 * (24 + 42)
= 4 * 66
= 264
Therefore, the sum of the first 8 terms of the A.P. is 264.
To solve this problem, we need to find the value of the first term (a) and the common difference (d).
Let's say the first term is 'a' and the common difference is 'd'. The 5th term will be given by the formula:
5th term = a + (5-1)d = a + 4d
Given that the 5th term is 3 times the first term:
a + 4d = 3a
Now, we are also given that 1/4 of the 5th term is 9:
(1/4)*(a + 4d) = 9
We can simplify this equation by multiplying both sides by 4:
a + 4d = 36
Since we have two equations with two unknowns (a and d), we can solve them simultaneously.
First, let's solve the equations a + 4d = 3a and a + 4d = 36 by eliminating variable 'a'.
Subtracting the first equation from the second equation:
( a + 4d ) - ( 3a ) = 36 - 3a
-2a + 4d = 36 - 3a
Rearranging:
-2a + 3a + 4d = 36
a + 4d = 36
We found that this is the same as the second equation we had earlier. Therefore, we can say that the value of 'a' and 'd' can be any values that satisfy a + 4d = 36.
Now, let's find the sum of the first 8 terms of the AP.
The sum of n terms is given by the formula:
Sum = (n/2)(2a + (n-1)d)
In this case, we want to find the sum of the first 8 terms, so n = 8.
Sum = (8/2)(2a + (8-1)d)
Sum = 4(2a + 7d)
Sum = 8a + 28d
Since we have an equation with two variables (a and d) and only one equation (a + 4d = 36), there are infinite possible values for 'a' and 'd'. Therefore, we cannot find a unique value for the sum of the first 8 terms without additional information.