Question

what is the nth term of the following AP whose sum is 700
160,140,120....

Answer 1

here first term (a ) = 160
common difference (d ) = 140 -160 = -20
use formula,
Sn = n/2{ 2a + ( n -1) d}
700 =n/2{ 2×160 +( n -1)(-20)}
1400 =n( 320 -20n +20 )
1400 = 340n -20n²
70 = 17n -n²
n² -17n +70 =0
n² -10n -7n +70 =0
( n -10)(n -7) =0
n = 7, 10

here we see nth term = 7, and 10 both are possible , because
find 7th term = a + (7 -1)d = 160 -6(20) = 40
find 10th term = a + (10 -1)(-20) = 160 -180 = -20

7th term = 40
8th term = 20
9 th term = 00
10the term =-20
you see sum of 7th +8th +9th +10th = 7th term

hence , sum of 8th + 9th and 10th doesn't change the value of sum
so, n = 7 and 10 both are possible

Answer 2

its an textbook question