How many terms of the A.P. are needed to give the sum –25?
The given A.P. is − 6 , − 11 2 , − 5 , … .
Let a , d be the first term and common difference of the given A.P.
a = − 6 d = − 11 2 + 6 = 12 − 11 2 = 1 2
Let the sum of n terms of the given A.P. be − 25 .
The formula for the sum of n terms in an A.P. is given by,
S n = n 2 [ 2 a + ( n − 1 ) d ]
Substitute the values of a , d and S n as − 6 , 1 2 , − 25 in the above expression.
− 25 = n 2 [ 2 × ( − 6 ) + ( n − 1 ) × ( 1 2 ) ] − 25 × 2 = n [ − 12 + n 2 − 1 2 ] − 50 = n [ − 25 2 + n 2 ] − 50 × 2 = n ( − 25 + n )
Further simplify the above expression.
− 100 = n ( − 25 + n ) 100 = n ( n − 25 ) 100 = n 2 − 25 n n 2 − 25 n − 100 = 0
Further simplify the above equation.
n 2 − 25 n − 100 = 0 n 2 − 5 n − 20 n − 100 = 0 n ( n − 5 ) − 20 ( n − 5 ) = 0 ( n − 20 ) ( n − 5 ) = 0
Equate the above expression to obtain the value of n .
n = 20 or 5 .
Thus, the total number of terms in the A.P. − 6 , − 11 2 , − 5 , … is 20 or 5 .
The given A.P. is
Let
Let the sum of
The formula for the sum of
Substitute the values of
Further simplify the above expression.
Further simplify the above equation.
Equate the above expression to obtain the value of
Thus, the total number of terms in the A.P.
