Question

Sum of
of
first six terms of AP is 6 . Product
2nd & 5th teams is - 80 find the terms.​

Answer 1

Answer:

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Answer 2

The First term of Arithmetic Progression is - 14

Step-by-step explanation:

Given as :

For Arithmetic Progression

The sum of first 6 term of an A.P = 6

For A.P , Sum of n terms = $S_n$

Or, $S_n$  = $\dfrac{n}{2}$  [ 2 a + ( n - 1 ) d ]                           where a is first term

                                                                         d is common difference

or n = 6    

    $S_6$  = $\dfrac{6}{2}$  [ 2 a + ( 6 - 1 ) d ]

Or, 6 = 3 [ 2 a + 5 d ]

Or, 2 a + 5 d = $\dfrac{6}{3}$

∴     2 a + 5 d = 2                          ............1

Again

product of 2nd term and 5th term =  -8

Now,

nth term of an A.P = $t_n$  = a + (n - 1)d

For n = 2

$t_2$  = a + (2 - 1)d

i.e  $t_2$  = a + d

And

For n = 5

$t_5$  = a + (5 - 1)d

i.e  $t_5$  = a + 4 d

∵   $t_2$ × $t_5$   = - 8

So,   (a + d) × (a + 4 d) = - 8                   .....2

From eq 1     ,   2 a + 5 d = 2    

or,      2 a = 2 - 5 d

∴           a = 1 - 2.5 d

Put the value of a in eq 2

 (1 - 2.5 d + d) × (1 - 2.5 d + 4 d) = - 80

Or,  (1 - 1.5 d)  × ( 1 + 1.5 d) = - 80

or,   1 - 2.25 d² = - 80

or,  2.25 d² = 81

∴             d = $\sqrt{\dfrac{81}{2.25} }$

              d = 6

So, The value of d = 6

Put the value of d in to eq 1

2 a + 5 × 6 = 2        

2 a + 30 = 2

Or,   2 a = - 28

Or,       a = - 14

So, The  First term of A.P = a = - 14

Hence, The  First term of Arithmetic Progression is - 14  . Answer