SOLUTION: The second and fifth terms are in a geometric progression are 12 and 40.5 respectively. Find the first term and the common ratio. Hence, write down an expression for the nth term.

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Question 1202406: The second and fifth terms are in a geometric progression are 12 and 40.5 respectively. Find the first term and the common ratio. Hence, write down an expression for the nth term.
Found 3 solutions by greenestamps, ikleyn, mananth:
Answer by greenestamps(13216) About Me  (Show Source):
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Use the standard convention of a for the first term and r for the common ratio.

First term: a
Second term: ar
Fifth term: ar%5E4

The ratio of the fifth and second terms is the common ratio, cubed. Use that to determine the common ratio.

r%5E3=40.5%2F12=81%2F24=27%2F8
r=3%2F2

The first term is the second term, divided by the common ratio.

a=12%2F%283%2F2%29=12%2A%282%2F3%29=8

ANSWERS:
1st term: 8
formula for n-th term (ar^(n-1): t%28n%29=8%28%283%2F2%29%5E%28n-1%29%29

Answer by ikleyn(52915) About Me  (Show Source):
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.
The second and fifth terms highlight%28cross%28are%29%29 in a geometric progression are 12 and 40.5 respectively.
Find the first term and the common ratio. Hence, write down an expression for the nth term.
~~~~~~~~~~~~~~~~~~~~~~ 

Write the terms in standard form

    a%5B2%5D = a*r,  a%5B5%5D = a%2Ar%5E4,


Take the ratio

    a%5B5%5D%2Fa%5B2%5D = %28a%2Ar%5E4%29%2F%28a%2Ar%29 = r%5E3 = 40.5%2F12 = 3.375.


It implies  r = root%283%2C3.375%29 = 1.5.


Thus the common ratio is  r= 1.5.


Now the first term is  a = a%5B2%5D%2Fr = 12%2F1.5 = 8.


The expression for the n-th term is  a%5Bn%5D = a%2Ar%5E%28n-1%29 = 8%2A1.5%5E%28n-1%29.

Solved.     //     All questions are answered.

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On geometric progressions,  see introductory lessons
    - Geometric progressions
    - The proofs of the formulas for geometric progressions
    - Problems on geometric progressions
    - Word problems on geometric progressions
in this site.

Learn the subject from there.



Answer by mananth(16946) About Me  (Show Source):
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The second and fifth terms are in a geometric progression are 12 and 40.5 respectively. Find the first term and the common ratio. Hence, write down an expression for the nth term.
nth term of a GP is an=ar^(n-1)
2nd term
a2 = a*r^(2-1)
12= a*r
5th term
40.5= a*r^(5-1)
40.5 = ar^4
40.5= ar*r^3
40.5 = 12*r^3 (ar=12)
40.5/12 = r^3
3.375=r^3
3 15/40=r^3
135/40 =r^3
27/8 =r^3
cube root
r= 3/2

ar= 12
r = 3/2
a*3/2=12
a=8
an = a*r^(n-1)
an = 8*3/2 ^(n-1)