Question

the 19th term of an ap is equal to 3times it's 6th term if it's 9th term is 19 find ap

Answer 1

Answer:

Step-by-step explanation:

a₁₉ = 3 ( a₆ )

⇒ a + 18d = 3 ( a + 5d )

⇒ a + 18d = 3a + 15d

⇒ 3a - a + 15d - 18d = 0

⇒ 2a - 3d = 0   → Equation 1

Also given that,

a₉ = 19

⇒ a + 8d = 19   → Equation 2

⇒ 2a + 16d = 38    ( Multiplying Equation 2 by 2 )

Subtracting Equations 1 and 2, we get,

⇒ 16d - ( -3d ) = 38 - 0

⇒ 19d = 38

⇒ d = 38/ 19 = 2

Hence Common difference is 2

Substituting this in Equation 1 we get,

⇒ 2a + 16 ( 2 ) = 38

⇒ 2a + 32 = 38

⇒ 2a = 38 - 32 = 6

⇒ 2a = 6

⇒ a = 6/2 = 3

Hence first term is 3.

⇒ AP = 3, 3+2, 3+2(2), 3+3(2), ...

⇒ AP= 3,5,7,9, ...

Hope it helped !

Answer 2

a9 = 19 => (a1 + 8d) = 19 => a1 = 19 – 8d
a19 = 3 × a6
(a1 + 18d) = 3 × (a1 + 5d)
(19 – 8d +18d) = 3 × (19 – 8d + 5d)
(19 + 10d) = 3 × (19 – 3d)
19 + 10d = 57 – 9d
19d = 19(4)
d = 4.
a1 = 19 – 8d = 19 – 8(4) = –13
Therefore, A.P. is,
–13, –9, –5, .....