Question

The 19 terms of an a.p is equal to three times it's second term .if it 9 term is 19.find the a.p

Answer 1

GIVEN :–

• 19th term of A.P. = Three time of second term

• 9th term of A.P. = 19

TO FIND :–

• A.P. = ?

SOLUTION :–

• We know that nth term –

$\\ \: \longrightarrow \: \large { \boxed{ \sf T_{n} \: = a + (n - 1)d}}$

• Here –

$\\ \: \: \: \: \: \: { \huge{.}} \: \: \: \sf T_{n} \: = n \th \: \: term$

$\\ \: \: \: \: \: \: { \huge{.}} \: \: \: \sf a \: = first \: \: term$

$\\ \: \: \: \: \: \: { \huge{.}} \: \: \: \sf d \: = common \: \: difference$

• According to the first condition –

$\\ \implies \sf 19 \th \: \: term \: \: of \: \: A.P. =3(second \: \: term)$

$\\ \implies \sf a + (19 - 1)d =3[a + (2 - 1)d]$

$\\ \implies \sf a + 18d =3(a +d)$

$\\ \implies \sf a + 18d =3a +3d$

$\\ \implies \sf 18d - 3d =3a - a$

$\\ \implies \sf 15d=2a$

$\\ \implies \sf a = \dfrac{15}{2}d \: \: \: \: \: - - -eq.(1)$

• According to the second condition –

$\\ \implies \sf 9 \th \: \: term \: \: of \: \: A.P. = 19$

$\\ \implies \sf a + ( 9 - 1)d = 19$

$\\ \implies \sf a + 8d = 19$

• Now Using eq.(1) –

$\\ \implies \sf \dfrac{15}{2}d + 8d = 19$

$\\ \implies \sf \left( \dfrac{15 + 16}{2} \right)d= 19$

$\\ \implies \sf \left( \dfrac{31}{2} \right)d= 19$

$\\ \implies \sf 31d= 38$

$\\ \implies \large { \boxed{ \sf d= \dfrac{38}{31} }}$

• Put the value of 'd' in eq.(1) –

$\\ \implies \sf a = \left( \dfrac{15}{2} \right) \left( \dfrac{38}{31} \right)$

$\\ \implies \large { \boxed{ \sf a= \dfrac{285}{31} }}$

• Hence , A.P. $\implies \sf \: \dfrac{285}{31} \: , \: \dfrac{323}{31} \: ,\: \dfrac{361}{31} ,......$

Answer 2

${\huge{\bf{\red{\underline{Solution:}}}}}$

${\bf{\blue{\underline{Given:}}}}$

• 19 term = 3times second term
• 9 term = 19

${\bf{\blue{\underline{To\:Find:}}}}$

• A.P =?

${\bf{\blue{\underline{Now:}}}}$

$: \implies{\sf{ a_{19} = 3 a_{2} }}$

$: \implies{\sf{ a + 18d = 3 (a + d) }}$

$: \implies{\sf{ a + 18d = 3 a + 3d }}$

$: \implies{\sf{ a -3a= 18d-3d}}$

$: \implies{\sf{ -2a=- 15d}}$

$: \implies{\sf{ 2a= 15d}}$

$: \implies{\sf{ a = \frac{15d}{2} .....(1)}}$

___________________________________

$: \implies{\sf{ a_{9} =19}}$

$: \implies{\sf{ a + 8d=19.....(2)}}$

Put value of a in (2)

$: \implies{\sf{ \frac{15d}{2} + 8d=19}}$

$: \implies{\sf{ \frac{15d + 16d}{2} =19}}$

$: \implies{\sf{ \frac{31d}{2} =19}}$

$: \implies{\sf{ 31d = 38}}$

$: \implies{\sf{ d= \frac{38}{31}}}$

___________________________________

$: \implies{\sf{ a = \frac{15}{2} } \times \frac{38}{31} }$

$: \implies{\sf{ a = \frac{570}{62} } }$

Hence the A.P a , a+d , a+2d.....

$: \implies{\sf{ \frac{285}{31} ,\frac{323}{31} , \frac{361}{31} } }$