Geometric and Arithmetic Progressions?

Question

The #1st, 2nd and 3rd# terms of a geometric progression are the #1st, 9th, and 21st# terms respectively of an arithmetic progression. The #1st# term of each progression is #8# and the common ratio of the geometric progression is #r#, where #r != 1#. Find
(i) the value of r,
(ii) the #4th# term of each progression.

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Answer 1 · 2018-05-31T19:03:13

# (i) : r=3/2, (ii) :" the "4^(th)" term of GP is "27" and that of AP is "25/2#.

The #1^(st)# term of the GP is #8#, and the common

ratio is #r!=1#.

Hence, the #2^(nd)# and the #3^(rd)# terms of the GP must be

#8r# and #8r^2#.

Now, it is given that these are #9^(th) and 21^(st)# terms of an AP, of

which, #1^(st)# term is #8#.

So, if #d# is the common difference of the AP, then, we have

#8r=8+(9-1)d, i.e., 8r=8+8d, or, r=1+d...(ast^1)#,

#8r^2=8+20d, or, 2r^2=2+5d...................(ast^2)#.

#"By "(ast^1)" then, "2r^2=2+5(r-1)=5r-3#.

#rArr 2r^2-5r+3=0#.

#rArr (r-1)(2r-3)=0#.

#:. r=1#, which contradicts the given that #r!=1, or, r=3/2#.

Therefore, the #4^(th)# of the GP is #ar^3=8*(3/2)^3=27#, and, that of the AP is #8+(4-1)d=8+(4-1)(3/2)=25/2#.