Question

The sum of the 6th and 15th elements of an arithmetic progression is equal to the sum of 7th, 10th and 12th elements of the same progression. Which element of the series should necessarily be equal to zero? (a) 10th (b) 8th (c) 1st (d) none of these

Answer 1

ANS :

OPTION B : 8TH TERM

⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕

THANKS FOR THE QUESTION !

⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕

=> T STANDS FOR TERM .

=> n STANDS FOR TERM NO.

=> A STANDS FOR FIRST TERM IN AN AP

=> D STANDS FOR DIFFERENCE BETWEEN TWO CONJUGATIVE TERMS .....

⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕

=> BY USING FORMULA :

=> Tn = A + D ( n - 1 )

⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕

6TH TERM :

=> T6 = A + D ( 6 - 1 )

=> T6 = A + 5D

⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕

15TH TERM :

=> T 15 = A + D ( 15 - 1 )

=> T 15 = A + 14D

⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕

7 TH TERM :

=> T7 = A + D ( 7-1 )

=> T7 = A + 6D

⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕

10TH TERM :


=> T 10 = A + D ( 10-1 )

=> T 10 = A + 9D

⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕

12 TH TERM

=> T 12 = A + D ( 12-1 )

=> T 12 = A + 11D

⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕

BY GIVEN CONDITION ,

=> T6 + T15 = T7 + T10 + T12

=> A + 5D + A + 14D = A + 6D + A + 9D + A + 11D

=> 2A + 19D = 3A + 26D

=> 2A - 3A = 26D - 19D

=> - A = 7D

=> A = -7D

⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕

NOW ,

BY GIVEN OPTIONS :

=> 8TH TERM MUST BE 0

BECAUSE ,

⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕

8TH TERM :

=> T8 = A + D ( 8 - 1 )

=> T8 = A + 7D ............A = - 7D

=> T8 = -7D + 7D

=> T8 = 0

⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕

HOPE IT WILL HELP U ........

THANKS AGAIN ........

⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕⭕

Answer 2

ANSWER

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

Then third term will be =a+2d

${15}^{th} \: will \: be = a + 14d$

${6}^{th} \: will \: be = a + 5d$

$1 {1}^{th} \: will \: be = a + 10d$

$1 {3}^{th} will \: be = a + 12d$

$then \: the \: eq. \: will \: be$

$a + 2d + a + 14d = a + 5d + a + 10d + a + 12d$

$= > 2a + 16d = 3a + 27d$

$= > a + 11d = 0$

$we \: understand \: a + 11d \: will \: be \: the \: 1 {2}^{th} \: term \: of \: arithmetic \: progression.$

$so, \: CORRECT \: answer \: is \: {\boxed {\pink{12}}}$

HOPE IT'S HELPS YOU ❣️