Question

the sum of 3rd and 7th terms of an A.P. is 54 and the sum of the 5th and 11th term is 84. find the A.P.

Answer 1

$\boxed{\boxed{\mathbf{7, 12, 17, 22...}}}$

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We know that,

Sum of 3 and 7 term = 54

Sum of 5 and 11 term = 84

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Using the Formula,

$\mathsf{a_n = a + (n-1)d}$

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$\mathsf{a_3 = a + (3-1)d} \\ \bf\mathsf{=> a_3 = a + 2d -----(1)}$

Similarily,

$\bf\mathsf{=> a_7 = a + 6d -----(2)}$

$\bf\mathsf{=> a_5 = a + 4d -----(3)}$

$\bf\mathsf{=> a_{11} = a + 10d -----(4)}$

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Now,

Adding equation (1) and (2)

and equations (3) and (4)..... We get,

2a + 8d = 54

=> 2(a + 4d) = 54

=> $\bf\mathsf{a + 4d = 27}$ ______(5)

Also,

2a + 14d = 84

=> 2(a + 7d) = 84

=> $\bf\mathsf{a + 7d = 42}$ _______(6)

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Now,

$\underline{\textsf{By using Elimination Method}}$

Subtracting equation (6) by equation (5), we get,

=> $\mathsf{3d = 15}$

=> $\mathsf{d = \frac{15}{3}}$

=> $\boxed{\mathbf{d = 5}}$

Now, substituting value of d in equation (5) we get,

=> $\mathsf{a + 4(5) = 27}$

=> $\mathsf{a + 20 = 27}$

=> $\boxed{\mathbf{a = 7}}$

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First term of A.P = a = 7

Second Term of A.P = a + d = 7 + 5 = 12

Third Term of A.P = a + 2d = 7 + 5(2) = 7+10 = 17

Fourth Term of A.P = a + 3d = a + 3d = 7 + 5(3) = 7+ 15 = 22

Hence, the A.P is 7, 12, 17, 22......

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# Be Brainly

Answer 2

Let the first term of the AP be a and common difference be d,

                 According to the question : -

sum of 3rd and 7th term = 54

3rd term + 7th term = 54

From the identities of AP, $a_{x}=a+(x-1)d$

a + ( 3 - 1 )d + a + ( 7 - 1 )d = 54

a + 2d + a + 6d = 54

a + a + 2d + 6d = 54

2a + 8d = 54

2( a + 4d ) = 54

a + 4d = 27

a = 27 - 4d            ...( i )

sum of 5th and 11th term = 84

5th term + 11th term = 84

a + ( 5 - 1 ) d + a + ( 11 - 1 ) d =84

a + 4d + a + 10d = 8 4

a + a + 4d + 10d = 84

2a + 14d = 84

2( a + 7d ) = 84

a + 7d = 42

a = 42 - 7d          ...( ii )

           Comparing ( i ) and ( ii ) : -

42 - 7d = 27 - 4d

42 - 27 = 7d - 4d

15 = 3d

15 / 3 = d

5 = d { common difference }

          Substituting the value of d in ( i )

a = 27 - 4d

a = 27 - 4( 5 )

a = 27  - 20

a = 7

Therefore,

AP = a , a + d , a + 2d , a + 3d...........∞

     = 7 , 7 + 5 , 7 + 10 , 7 + 15 ............∞

     = 7 , 12 , 17 , 22 ............∞