the fourth term of an ap is 11 And the eight term exceeds twice the fourth term by 5.find the ap and sum of first 20 terms.
T4 = 11
T8 = 2*11+5 = 27
so, T8-T4 = 4d = 16
d = 4
a+3d = 11, so a = -1
S20 = 20/2 (2a+19d) = ...
To find the arithmetic progression (AP), we need to determine the first term (a) and the common difference (d).
Let's assume the first term is 'a' and the common difference is 'd'.
Given:
The fourth term is 11, which means 𝑎 + 3𝑑 = 11 ---(1)
The eighth term exceeds twice the fourth term by 5, which means 𝑎 + 7𝑑 = 2(𝑎 + 3𝑑) + 5 ---(2)
To solve these two equations, we can use the method of substitution or elimination.
First, let's solve the equations using substitution:
From equation (1), we can express 'a' as a function of 'd':
𝑎 = 11 - 3𝑑
Substitute this value of 'a' in equation (2):
(11 - 3𝑑) + 7𝑑 = 2((11 - 3𝑑) + 3𝑑) + 5
Simplifying the equation:
11 - 3𝑑 + 7𝑑 = 22 - 6𝑑 + 5
Combining like terms:
4𝑑 + 11 = 27 - 6𝑑
Rearranging the equation:
10𝑑 = 16
𝑑 = 16/10
𝑑 = 1.6
Now, substitute the value of 'd' back into equation (1) to find 'a':
𝑎 + 3(1.6) = 11
𝑎 + 4.8 = 11
𝑎 = 11 - 4.8
𝑎 = 6.2
Therefore, the first term (𝑎) is 6.2 and the common difference (𝑑) is 1.6.
The AP is: 6.2, 7.8, 9.4, 11, 12.6, 14.2, 15.8, 17.4, ...
To find the sum of the first 20 terms of this AP, we can use the formula for the sum of an AP:
Sum = (n/2) * (2a + (n - 1) * d)
Where:
n = number of terms
a = first term
d = common difference
Using this formula, substitute the following values:
n = 20
a = 6.2
d = 1.6
Sum = (20/2) * (2 * 6.2 + (20 - 1) * 1.6)
Simplify the equation:
Sum = 10 * (12.4 + 19 * 1.6)
Sum = 10 * (12.4 + 30.4)
Sum = 10 * 42.8
Sum = 428
Hence, the sum of the first 20 terms of the given AP is 428.