Question

The third term of an arithmetic series is -10 and the sum of the first eight terms of the series is 16.
Find the first term and common difference.

Answer 1

Answer:

First term = -26  and    Common difference = 8

Step-by-step explanation:

The general form of an Arithmetic Progression:

a, a + d, a + 2d, a + 3d,.............................

nth term of an AP series is Tn = a + (n - 1) d,

The third term of an arithmetic series  is -10.

( t3 ) :  a + 2d = -10........(1)

The sum of the first eight terms of the series is 16.

a + ( a + d ) + ( a + 2d ) + (a + 3d ) +(a + 4d ) +(a + 5d ) +(a + 6d ) +(a + 7d ) =16

8a + 28d = 16 .......(2)

From eqns (1) and (2).

a + 2d = -10.........(1)  x 8  = 8a + 16d = -80

8a + 28d = 16 ....(2)  x1    =8a + 28d = 16

eqns (1) - (2) Gives.

-12d = - 96

d =  -96 / -12

d = 8

sustituting "d" value in eqn (1)    

a + 2d = -10

a + 2(8) = -10

a + 16 = -10

a = -10 -16

a = -26

∴  First term = -26  and

Common difference = 8

Answer 2

Solution:-

Given:-

a3 =a + 2d = -10 -----------------(1)

S16 = 8/2[2a + (7)d] =16---------(2)

To Find:-

First term = a = ?

Common Difference = d = ?

Find:-

=)a3 = -10

=) a + 2d = -10

=) a = -10 - 2d--------(3)

Putting [ a = ( -10-2d) ] in eq (2).

=) 16 = 4 [ 2(-10-2d) + 7d]

=) 16 = 4 [- 20 - 4d + 7d ]

=) 16/4 = [ -20 + 3d ]

=) 4 = - 20 + 3d

=) 3d = 24

=) d = 8

Now,

Putting [ d = 8] in eq (3). we get,

=) a = -10 -2d

=) a = -10 - 2×8

=) a = -10 - 16

=) a = -26

Hence,

a = -26 and d = 8.