in an AP, if the 5th and 12th terms are 30 and 65 respectively. what is the sum of first 20terms
i dont no answer
To find the sum of the first 20 terms of an arithmetic progression (AP), we need to know the common difference between the terms.
In an AP, each term is obtained by adding the common difference to the preceding term. Let's assume the common difference is denoted by 'd.'
Given that the 5th term is 30, we can write the equation:
a + 4d = 30 *Equation 1*, where 'a' is the first term of the AP.
Similarly, the 12th term is 65, so we can write the equation:
a + 11d = 65 *Equation 2*.
To find the value of 'a' and 'd', we can solve these two equations simultaneously.
Subtracting *Equation 1* from *Equation 2*, we get:
7d = 35
d = 5
Substituting the value of 'd' back into *Equation 1*, we find:
a + 4(5) = 30
a + 20 = 30
a = 10
So, the first term of the AP, 'a', is 10, and the common difference, 'd', is 5.
To find the sum of the first 20 terms, we can use the formula for the sum of an AP:
Sn = (n/2)(2a + (n-1)d), where 'n' is the number of terms, 'a' is the first term, and 'd' is the common difference.
Substituting the values, we have:
S20 = (20/2)(2(10) + (20-1)(5))
S20 = 10(20 + 19(5))
S20 = 10(20 + 95)
S20 = 10(115)
S20 = 1150
Therefore, the sum of the first 20 terms of the AP is 1150.