The fourth term of an AP is 11 and the eighth term exceeds twice the fourth term by 5. Find the AP and the sum of first 20 terms
a+3d=11
a+7d=2*11+5
Solve for a and d, and then, of course,
S20 = 10(2a+19d)
To solve this problem, we need to find the common difference (d) of the arithmetic progression (AP) and then form the AP based on the given information.
Let's start by finding the common difference:
The fourth term is given as 11, so we can write the equation:
a + 3d = 11 ...(1) (where a is the first term)
The eighth term exceeds twice the fourth term by 5:
a + 7d = 2(11) + 5 = 27 ...(2)
Now, we have a system of two equations (1) and (2) with two variables (a and d). We can solve this system to find the values of a and d.
Subtracting equation (1) from equation (2), we get:
(a + 7d) - (a + 3d) = 27 - 11
4d = 16
d = 4
Substituting the value of d back into equation (1), we get:
a + 3(4) = 11
a + 12 = 11
a = 11 - 12
a = -1
The first term (a) is -1 and the common difference (d) is 4. Therefore, the AP is: -1, 3, 7, 11, ...
To find the sum of the first 20 terms of the AP, we can use the formula:
Sn = (n/2)(2a + (n-1)d)
Substituting the values, we have:
S20 = (20/2)(2(-1) + (20-1)(4))
= 10(-2 + 19(4))
= 10(-2 + 76)
= 10(74)
= 740
Therefore, the sum of the first 20 terms of the AP is 740.