Question

the 19th term of an ap is equal to three times its 6th term if the 9th term of the AP is 19 find the first term and the common difference

Answer 1

Heya !!!




9th term = 19


A + 8D = 19



A = 19 -8D -------(1)


19th term = 3 ( 6th term )



A + 18D = 3 ( A + 5D)



A + 18D = 3A + 5D



3A - A = 18 D - 5D -----(2)


Putting the value of A in equation (2)


3 × ( 19-8D ) - ( 19 - 8D) = 18D - 5D




57 - 24D - 19 + 8D = 13D



- 16D + 38 = 13D



-16D - 13D = -38



-19D = -38


D = -38/-19



D = 2.



Putting the Value of D in EQUATION (1)



A = 19 -8D = 19 - 8 × 2




A = 19 -16


A = 3.




Hence,


First term (A) = 3 and Common Difference (D) = 2.




HOPE IT WILL HELP YOU..... ::)

Answer 2

Hi,

Here is your answer,

19th term of an A.P       We know that, a = First term , d = common                                                                                                                            difference 
                                       ∴   3 times its 6th term.
 → a + 18d 

→ 3 (a + 5d )

→ a + 18d = 3a + 15 d         ∴ ( we will take 18d and 15d as 1 term and a and 3a and another term )

      → 2a = 3d
      
        → a = 3d/2       ------------. (1)

Now, 

     →   T₉ = 19 = a + 8d = 19    

Substitute the a value in the given equation

→ 3d/2 + 8d = 19         ----------> from (1)

→ 3d + 16d/2 = 19

→ 19d = 19 × 2 

→ 19d = 36

→ d = 36/19

→ d = 2

Now, Substitute the d = 2 value in equation (1)

 → a = 3d/2 

→ a = 3 ×2 /2 = 6/2 = 3

a = 3 and d = 2 

Hope it helps you !