Question

How many terms of the series 54,51, 48, ..........be taken so that their sum is 513 ?

Answer 1

Use forumla ,
Sn=n/2{2a+(n-1)d}

from series ,
a=54
d=-3
Sn=513
now,
513=n/2{54 x 2+(n-1)(-3)}
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