Question

in an AP, if the 5th and 12th terms are 30 and 65 respectively. what is the sum of first 20 terms of the A.P

Answer 1

a+4d=30
a+11d=65
Therefore
7d=35
d=5
therefore
a+20=30
a=10
sum =10×[20+(20-1)5]
=10×115
=1150

Answer 2

In general,

Nth term of A.P = a + (n-1)d,
5th term of A.P = a + 4d = 30 ........(1),

12h term of A.P = a + 11d = 65......(2),

Solving (1) and (2)
7d = 35,
=> d = 5,

=> a + 4(5) = 30,
=> a + 20 = 30,
=> a = 10,

Sum of n terms of A.P = (n/2) *(2a+(n-1)d),

Sum of first 20 terms = 10*(20+19*5)
=> 10(115)
=> 1150,

Therefore the sum of first 20 numbers of A.P is 1150,

Hope you understand ,

Have a great day !!