Question

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Answer 1

Let,

a = first term

d = common difference

Acc. to the given 1st condition,

t3 + t7 = 6

a + 2d + a + 6d = 6

2a + 8d = 6

Dividing by 2 on both sides

a + 4d = 3.............. ( i )

Acc. to the given 2nd condition,

t3*t7 = 8

(a + 2d) (a + 6d) = 8

a2 + 12d2 = 8.......... (ii)

Squaring eqn (i) on both sides

a2 + 16d2 = 9.......... (iii)

Subtracting eqn (ii) and (iii)

4d2 = 1

d2 = 1/4

Taking square root on both sides

d = 1/2

Put d =1/2 in eqn (i)

a + 4d = 3

a + 4*1/2 = 3

a + 2 = 3

a = 3-2

a = 1

For S16

a = 1 and d = 1/2

n= 16

$sn \: = \frac{n}{2} (2a + (n - 1)d)$

Sn = 16/2 ( 2*1 + 15*1/2)

Sn = 8 (2+7.5)

Sn = 8*9.5

Sn = 76
Hope it helps you out...

Answer 2

Answer:

S16 = 76 or S16 = 20

Step-by-step explanation:

For an A.P. tn = a + ( n-1 ) x d

According to the first condition

t3 + t7 = 6

a + 2d + a + 6d = 6

2a + 8d = 6

Divide throughout by 2 and you get

a + 4d = 3

Therefore    a = 3 - 4d

According to the second condition  

( a + 2d ) ( a + 6d ) = 8

Substitute a = 3 - 4d in the above equation and you get

( 3 - 4d + 2d ) ( 3 - 4d + 6d ) = 8

( 3 - 2d ) ( 3 + 2d ) = 8

Therefore by using the identity  a² - b² = ( a + b ) ( a - b ) you get

( 3 ) ² - ( 2d )² = 8

9 - 4d² = 8

9 - 8 = 4d²

1/4 = d²

Therefore d = +1/2 = 0.5    or d = -1/2 = -0.5

If d = +0.5 then a = 3 - 4d    

                           = 3 - 2

                           = 1

If d = -0.5 then a = 3 - 4d

                           = 3 - ( - 2 )

                          = 5

For a = 1 and d = 0.5 the 16th term is

=a + 15d

= 1 + 15 x 0.5

= 1 + 7.5

= 8.5

So Sn = n/2 ( a + tn )    

S16 = 16/2 ( 1 + 8.5 )

S16 = 8 x 9.5

S16 = 76

For a = 5 and d = -0.5 the 16th term is

= a + 15d

= 5 + ( - 7.5 )

= 5 - 7.5

= -2.5

So Sn = n/2 ( a + tn )    

S16 = 16/2 ( 5 - 2.5 )

S16 = 8 x 2.5

S16 = 20

Therefore S16 = 76 or S16 = 20

Hope this helps and mark me as brainliest if my answer helped you.