Question

which term of A.P 3,15, 27,39..... will be 120 more than it's 21 term.​

Answer 1

${\fbox{\boxed {\huge{\rm{\blue{Answer}}}}}}$

Step-by-step explanation:

a=3 , d=15-3=12 , n=21

an= a + (n - 1) d

an = 3 + (21 - 1) (12)

an= 3 + (20) (12)

an = 240 + 3

an= 243

an= 243 + 120

an= 363

363= 3 + (n - 1) (12)

363 = 3+ 12n - 12

363 - 3 = 12n - 12

360 + 12 = 12n

372 = 12n

n = 372/12

n = 31

${\boxed{\huge{\green{\mathcal{BeBrainly}}}}}$

Answer 2

QUESTION:-

ᴡʜɪᴄʜ ᴛᴇʀᴍ ᴏғ ᴀ.ᴘ 3,15, 27,39..... ᴡɪʟʟ ʙᴇ 120 ᴍᴏʀᴇ ᴛʜᴀɴ ɪᴛ's 21 ᴛᴇʀᴍ.

ANSWER✓

$\Large\bold\purple{given,}$

$\sf\dashrightarrow a=3$

$\sf\dashrightarrow t_1=3$

$\sf\dashrightarrow t_2=15$

$\sf\dashrightarrow d=t_2 -t_1$

$\sf\dashrightarrow d=15-3=12$

$\sf\dashrightarrow n=21$

$\sf\dashrightarrow term\:of\:A.P\:will\:be\:120\:more\:than\:term\:21th\:term$

TO FIND,

$\sf\large\dashrightarrow the\:value\:of\:n$

$\Large\underline\bold{SOLUTION,}$

$\sf\dashrightarrow to\: understand\:it\:we\:will\:take\:2\:cases$

$\sf\therefore in\:case\:1\:we\:will\:find\:t_n\:of\:21st\:term$

$\sf\therefore in\:2\:case\:we\:will\:find\:the\:term\:of\:ap\:of\:120\:more\:than\:21st\:term$

$\sf\large\underline\therefore ATQ...i.e.,.... according\:to\:the\:question$

$\large {\fbox {CASE:-1}}$ $\sf\implies t_n\:of\:the\:21st\:term$

$\sf\underline\therefore n^{th}\:term\:t_n\:is\:given\:by,$

$\large{\boxed{\sf{t_n=a+(n-1) \times d}}}$

$\sf\implies t_n=3+(21-1) \times 12$

$\sf\implies t_n=3+(20) \times 12$

$\sf\implies t_n=3+240$

$\sf\implies t_n=243$

$\sf{\boxed{\sf{t_n=243}}}$

$\sf\therefore AP\:will\:be\:120\:more\:than\:it's\:21_st\:term$

then,

$\sf\therefore AP= 243+120=363$

$\large {\fbox {CASE:-2}}$

$\sf\bold\purple{given,}$

$\sf\therefore t_n=363$

$\sf\therefore a=3$

$\sf\therefore d=12$

$\sf\bold\red{to\:find,}$

$\sf\implies nth\:term\:of\:the\:t_n\:363$

$\sf\underline\therefore n^{th}\:term\:t_n\:is\:given\:by,$

$\large{\boxed{\sf{t_n=a+(n-1) \times d}}}$

$\sf\implies 363=3+(n-1) \times 12$

$\sf\implies 363=3+(12n-12)$

$\sf\implies 363-3=12n-12$

$\sf\implies 360+12=12n$

$\sf\implies 372=12n$

$\sf\implies \dfrac{372}{12} =n$

$\sf\implies n=\cancel \dfrac{372}{12}$

$\sf\implies n=31$

$\large{\boxed{\sf{n=31}}}$

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ADDITIONAL NOTES,

• those sequences whose terms follow certain pattern are called

❝progressions❞

• there are 4 types of progressions

1. arithmetic progression (A.P)
2. geometric progression (G.P.)
3. harmonic progression (H.P.)
4. arithmetic-geometric progression(A.G.P)

• sum of first n terms is given by

$\sf\therefore s_n= \dfrac{n}{2} \bigg[ 2a+(n-1) \times d \bigg]$

$\sf\therefore \dfrac{n}{2} \bigg[ a+l \bigg]$

• ARITHMETIC MEAN,

$\sf\therefore A-a=b-A$

$\sf\implies 2A=a+b$

$\large{\boxed{\sf{A= \dfrac{a+b}{2}}}}$

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