AP/GP question

**margaritas** · September 2nd 2006, 07:12 PM

Here goes:

The first 3 terms of a geometric progression, with first term a = 1 and common ratio r, r does not equal 1, are the 1st, 2nd and 4th term of an arithmetic progression. Find the value of r. (Answer: r = 2)

I got 3 as my answer though. Thanks in advance if you could help solve this question!

**shubhadeep** · September 2nd 2006, 08:16 PM

Originally Posted by margaritas

Here goes:

The first 3 terms of a geometric progression, with first term a = 1 and common ratio r, r does not equal 1, are the 1st, 2nd and 4th term of an arithmetic progression. Find the value of r. (Answer: r = 2)

I got 3 as my answer though. Thanks in advance if you could help solve this question!

For the G.P.,
$a = 1$
$common$ $ratio = r (r$ $is$ $not$ $equal$ $to$ $1)$
$a_2=r$
$a_3=r^{2}$

Now since the first three terms of the G.P. are the first, second and fourth terms of an A.P., therefore for the A.P.
$t = 1$
$t_2=r$
$t_4=r^{2}$
$common$ $difference,d=t_2-t=r-1$

$Since$ $fourth$ $term$ $of$ $the$ $A.P.$ $is$ $the$ $third$ $term$ $of$ $the$ $G.P.$
$a_3=t_4=t+3d$
$r^{2}=1+3(r-1)$
$r^{2}=1+3r-3$
$r^{2}-3r+2=0$
$(r-2)(r-1)=0$
$r=1,r=2$
$But$ $since$ $r$ $is$ $not$ $equal$ $to$ $1(given),$ $hence$ $\boxed{r=2}.$

I got 3 as my answer though.

I think you probably made a mistake in factorisation.

**margaritas** · September 2nd 2006, 08:56 PM

Oh so the 'a' for AP is assumed to be 1 as well?

EDIT: Thanks for the solution, shubhadeep!

**shubhadeep** · September 2nd 2006, 09:39 PM

Originally Posted by margaritas

Oh so the 'a' for AP is assumed to be 1 as well?

Its not an assumption. Your question clearly states that.

Originally Posted by margaritas

The first 3 terms of a geometric progression, with first term a = 1 and common ratio r, r does not equal 1, are the 1st, 2nd and 4th term of an arithmetic progression.

The first term of the G.P. is the first term of the A.P., the second term of the G.P. is the second term of the A.P. and the third term of the G.P. is the fourth term of the A.P.

**margaritas** · September 3rd 2006, 01:30 AM

Oh ok, silly me.

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