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Question Number 12304 by Gaurav3651 last updated on 18/Apr/17

H o w m a n y g e o m e t r i c p r o g r e s s i o n s

i s / a r e p o s s i b l e c o n t a u n i n g 27, 8

a n d 12 a s t h r e e o f i t s / t h e i r t e r m s ?

(a) 1

(b) 2

(c) 4

(d) i n f i n i t e l y m a n y

Answered by mrW1 last updated on 18/Apr/17

{l e t}^{'} s a s s u m e 8, 12 a n d 27 a r e 3 t e r m s o f

a G P w i t h t h e c o m m o n r a t i o q :

8 \dots (n - 1 t e r m s) \dots 12 \dots . (m - 1 t e r m s) \dots 27

\frac{27}{12} = \frac{3 \times 3 \times 3}{3 \times 2 \times 2} = \frac{3 \times 3}{2 \times 2} = q^{m} (i)

\frac{12}{8} = \frac{2 \times 2 \times 3}{2 \times 2 \times 2} = \frac{3}{2} = q^{n} (i i)

q^{m - n} = \frac{3}{2} (i) / (i i)

\Rightarrow q = {(\frac{3}{2})}^{\frac{1}{m - n}}

q^{n} = {(\frac{3}{2})}^{\frac{n}{m - n}} = \frac{3}{2} f r o m (i i)

\Rightarrow \frac{n}{m - n} = 1

n = m - n

\Rightarrow m = 2 n

\Rightarrow q = {(\frac{3}{2})}^{\frac{1}{n}} f r o m (i i)

i . e . f o r a n y n ⩾ 1 w e c a n a l w a y s f i n d

a G P w h o s e 1 s t t e r m i s 8, (1 + n) - t h

t e r m i s 12 a n d (1 + 3 n) - t h t e r m i s 27,

a n d t h e i r c o m m o n r a t i o i s q = {(\frac{3}{2})}^{\frac{1}{n}} :

T_{1} = 8

\dots . . (n - 1 t e r m s b e t w e e n)

T_{1 + n} = 8 \times q^{n} = 8 \times {(\frac{3}{2})}^{\frac{n}{n}} = 12

\dots . . (2 n - 1 t e r m s b e t w e e n)

T_{1 + 3 n} = 8 \times q^{3 n} = 8 \times {(\frac{3}{2})}^{\frac{3 n}{n}} = 8 \times {(\frac{3}{2})}^{3} = 27

s i n c e a n y n ⩾ 1 f u l f i l l s t h i s, w e h a v e

i n f i n i t e l y m a n y s u c h G P .

\Rightarrow A n s w e r (d) i s c o r r e c t .

E x a m p l e s

n = 1 :

8, 12, 18, 27 \dots . .

n = 2 :

8, 8 \times {(\frac{3}{2})}^{\frac{1}{2}}, 12, 12 \times {(\frac{3}{2})}^{\frac{1}{2}}, 18, 18 \times {(\frac{3}{2})}^{\frac{1}{2}}, 27 \dots .

n = 3 :

8, 8 \times {(\frac{3}{2})}^{\frac{1}{3}}, 8 \times {(\frac{3}{2})}^{\frac{2}{3}}, 12, 12 \times {(\frac{3}{2})}^{\frac{1}{3}}, 12 \times {(\frac{3}{2})}^{\frac{2}{3}}, 18, 18 \times {(\frac{3}{2})}^{\frac{1}{3}}, 18 \times {(\frac{3}{2})}^{\frac{2}{3}}, 27 \dots .

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