Q-How-many-common-three-digit-numbers-are-there-in-the-following-two-sequences-a-n-1-5-9-13-b-m-4-7-10-13-

Question Number 173902 by mnjuly1970 last updated on 20/Jul/22

Q : H o w m a n y c o m m o n t h r e e - d i g i t n u m b e r s

a r e t h e r e i n t h e f o l l o w i n g

t w o s e q u e n c e s ?

{\begin{cases} a_{n} = 1, 5, 9, 13, \dots \\ b_{m} = 4, 7, 10, 13, \dots \end{cases}

Answered by mr W last updated on 21/Jul/22

a_{n} = 1 + 4 (n - 1)

b_{m} = 4 + 3 (m - 1)

a_{n} = b_{m}

1 + 4 (n - 1) = 4 + 3 (m - 1)

4 (n - 1) = 3 m

\Rightarrow n - 1 = 3 k, m = 4 k

\Rightarrow a_{n} = 1 + 12 k = b_{m}

1 + 12 k ⩾ 100 \Rightarrow k ⩾ \frac{99}{12} ⩾ 8

1 + 12 k ⩽ 999 \Rightarrow k ⩽ \frac{998}{12} ⩽ 83

\Rightarrow 83 - 8 + 1 = 76 n u m b e r s w i t h 3 d i g i t s

a r e c o m m o n i n s e q u e n c e s a_{n} a n d b_{n} .

Commented by Rasheed.Sindhi last updated on 21/Jul/22

G r e a t s i r!

Commented by mnjuly1970 last updated on 21/Jul/22

t h a n k s a l o t s i r W

e x c e l l e n t a s a l w a y s

g o d k e e p y o u m a s t e r W .

Commented by Tawa11 last updated on 21/Jul/22

Great sir

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