Q.24. How many terms of the AP : 54, 51, 48, 45 ........ must be taken to make the sum 513? Explain the reason for the double answer.
1026=n{108+3-3n}
=111n-3n^2
3n^2-111n+1026=0
n^2-37n+342=0
use quadratic formula ,
n={37+_√(1369-1368)}/2
=(37+_1)/2=19 and 18
here we see n gain 2 value e.g 19 and 18
now, we check it
18th term=54+(17)(-3)=54-51=3
19th term =54-18 x 3=0
you also see 19th term is zero
so adding or no adding 19th term value of sum is always 513
so , n gain two values
=111n-3n^2
3n^2-111n+1026=0
n^2-37n+342=0
use quadratic formula ,
n={37+_√(1369-1368)}/2
=(37+_1)/2=19 and 18
here we see n gain 2 value e.g 19 and 18
now, we check it
18th term=54+(17)(-3)=54-51=3
19th term =54-18 x 3=0
you also see 19th term is zero
so adding or no adding 19th term value of sum is always 513
so , n gain two values
