Question

the 19 term of A.p is equal to 3 times its terms. if 9 term of A.p of is 19 . find A.p​

Answer 1

Step-by-step explanation:

Hey buddy let's solve this beautiful problem!!!!!

$given \\ t19 = 3(t9) \\ t9 = 19$

Ok, now let's try to frame equations.

$t19 = 3(19) = 57$

$a + 18d = 57.......(1)$

$a + 8d = 19............(2)$

So, let's solve (1)&(2)=>

$10d = 38 \\ d = 3.8$

$now \: substitute \: them \: in \: one \: of \: the \: equations$

$a + 8(3.8) = 19 \\ a = 19 - 30.4 \\ so \: a = ( - 11.4)$

$so \: your \: required \: a.p \: is \: \\ (a - d) \: and\: a \:and \: (a + d)$

$(a - d) = ( - 15.2) \\ (a + d) = ( - 7.6)$

$now \: we \: have \: completed \: this \: question \: \\ required \: terms \: = \\ ( - 15.2) \: \: ( - 11.4) \: \: ( - 7.6)$

Hope it helps uh!

Answer 2

$\underline\mathfrak{Given:-}$

• 19th term of Ap is equal to three times of second term.
• 9th term of Ap is 19

$\underline\mathfrak{To \: \: Find:-}$

• Find the n .....?

$\underline\mathfrak{Solutions:-}$

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

• a = first term
• d = common difference
• nth = number of term
• an = last term

✰ 19th term of Ap is equal to three times of second term.

$\: \: \: \: \: \leadsto \: \: {a} \: + \: {({19} \: - \: {1})} \: d \: \: = \: \: {3} \: {[{a} \: + \: {({2} \: - \: {1})} \: d]}$

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

$\: \: \: \: \: \leadsto \: \: {a} \: + \: {18d} \: \: = \: \: {3a} \: + \: {3d}$

$\: \: \: \: \: \leadsto \: \: {18d} \: - \: {3d} \: \: = \: \: {3a} \: - \: {a}$

$\: \: \: \: \: \leadsto \: \: {15d} \: \: = \: \: {2a}$

$\: \: \: \: \: \leadsto \: \: {d} \: \: = \: \: \frac{2a}{15} \: \: \: \: \: ....{(i)}.$

✰ 9th term of Ap is 19.

$\: \: \: \: \: \leadsto \: \: {a} \: + \: {({9} \: - \: {1})} \: d \: \: = \: \: {19}$

$\: \: \: \: \: \leadsto \: \: {a} \: + \: {8d} \: \: = \: \: {19} \: \: \: \: \: ....{(ii)}.$

»★ Solving the Eq. (i) and Eq. (ii).

$\: \: \: \: \: \leadsto \: \: {a} \: + \: {8d} \: \: = \: \: {19}$

$\: \: \: \: \: \leadsto \: \: {a} \: + \: {8} \: \times \: \frac{2a}{15} \: \: = \: \: {19}$

$\: \: \: \: \: \leadsto \: \: {a} \: + \: \frac{16a}{15} \: \: = \: \: {19}$

$\: \: \: \: \: \leadsto \: \: {(\frac{{15} \: + \: {16}}{15})} \: {a} \: \: = \: \: {19}$

$\: \: \: \: \: \leadsto \: \: \frac{31a}{15} \: \: = \: \: {19}$

$\: \: \: \: \: \leadsto \: \: {31a} \: \: = \: \: {19} \: \times \: {15}$

$\: \: \: \: \: \leadsto \: \: {31a} \: \: = \: \: {285}$

$\: \: \: \: \: \leadsto \: \: {a} \: \: = \: \: \frac{285}{31}$

»★ Putting the value of a in Eq. (i).

$\: \: \: \: \: \leadsto \: \: {d} \: \: = \: \: \frac{2a}{15}$

$\: \: \: \: \: \leadsto \: \: {d} \: \: = \: \: \frac{2}{15} \: \times \: \frac{285}{31}$

$\: \: \: \: \: \leadsto \: \: {d} \: \: = \: \: \frac{38}{31}$

»★ Now, the Ap a, a + d and a + 2d

• $\: \: \: \: \: {a} \: \: = \: \: \frac{285}{31}$

• $\: \: \: \: \: {a} \: + \: {d} \: \: = \: \: \frac{285}{31} \: + \: \frac{38}{31} \: \: = \: \: \frac{{285} \: + \: {38}}{31} \: \: = \: \: \frac{285}{31}$

• $\: \: \: \: \: {a} \: + \: {2d} \: \: = \: \: \frac{285}{31} \: + \: {2} \: \times \: \frac{38}{31} \: \: = \: \: \frac{285}{31} \: + \: \frac{76}{31} \: \: = \: \: \frac{{285} \: + \: {76}}{31} \: \: = \: \: \frac{361}{31}$

»★ Hence,

$\: \: \: \: \: Ap \: \: is \: \: \frac{285}{31}, \: \: \frac{323}{31}, \: \: \frac{361}{31}$

$\underline\mathfrak{Important \: \: formula:-}$

• $\: \: \: \: \: {A_n} \: \leadsto \: a \: + \: {({n} \: - {1})} \: d$

$\underline\mathfrak{more \: \: information:-}$

• $\: \: \: \: \: \leadsto \: a \: \: = \: \: first \: \: terms$

• $\: \: \: \: \: \leadsto \: d \: \: = \: \: common \: \: difference$

• $\: \: \: \: \: \leadsto \: n \: \: = \: \: number \: \: of \: \: terms.$

• $\: \: \: \: \: \leadsto \: {a_n} \: \: = \: \: nth \: \: term$

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