The fourth term of an AP is 10 and eleventh term of it exceeds three times the fourth term by 1. Find the sum of first 20 terms.
find a and d using what they have told you:
a+3d = 10
a+10d = 3(a+3d)+1
Then just evaluate
S20 = 20/2 (2a+19d)
To find the sum of the first 20 terms of an arithmetic progression (AP), we need to know the first term and the common difference. However, we have only been given the fourth term and some information about the eleventh term.
Let's break down the given information step by step:
1. The fourth term of the AP is 10.
2. The eleventh term of the AP exceeds three times the fourth term by 1.
Using these two pieces of information, we can find the common difference (d) and the first term (a) of the AP.
Step 1: Finding the common difference (d):
We know that the fourth term is 10. In an arithmetic progression, the nth term can be represented as:
a + (n-1)*d
where a is the first term and d is the common difference.
Using the formula, we substitute n = 4 and the given fourth term = 10:
a + (4-1)*d = 10
a + 3d = 10 ...(Equation 1)
Step 2: Relating the eleventh term to the fourth term:
We are given that the eleventh term exceeds three times the fourth term by 1. In other words, the eleventh term can be expressed as:
a + (11-1)*d = 3*(a + (4-1)*d) + 1
a + 10d = 3(a + 3d) + 1
a + 10d = 3a + 9d + 1
10d - 9d = 3a - a + 1
d = 1 + a ...(Equation 2)
Step 3: Solving the simultaneous equations (Equations 1 and 2):
Substitute Equation 2 into Equation 1:
a + 3(1 + a) = 10
a + 3 + 3a = 10
4a + 3 = 10
4a = 10 - 3
4a = 7
a = 7/4
Now that we have found the value of a (the first term) and d (the common difference), we can find the sum of the first 20 terms of the AP.
Step 4: Finding the sum of the first 20 terms:
The sum of the first 20 terms (S20) of an AP can be calculated using the formula:
S20 = (n/2)*(2a + (n-1)d)
Substitute n = 20, a = 7/4, and d = 1 into the formula:
S20 = (20/2)*(2*(7/4) + (20-1)*1)
S20 = 10*(7/2 + 19)
S20 = 10*(7/2 + 38/2)
S20 = 10*(45/2)
S20 = 10*45/2
S20 = (10*45)/2
S20 = 450/2
S20 = 225
Therefore, the sum of the first 20 terms of the given arithmetic progression is 225.