Question

If 6th term of a G.P. is 46875 and its 4th term is 375. Find its 9th term.

Answer 1

let the first term =a ,
common ratio = r in GP
nth term = an = ar^n-1
a6 =46875 ⇒ar^5 = 46875----(1)

a4 =375⇒ar³ =375---(2)
do (1) /(2)
(ar^5)/(ar³) = 46875/375

a²=125
a= 5√5


a9 = ar^8
=ar^5 * ar³/a

= 46875 *375/5√5
= 3515625/√5
=703125√5

Answer 2

Formula we will be using:
(i) $a_n=ar^{n-1}\text{, where }a_n=$nth term, a=1st term and r=common ratio.
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Let the 1st term be 'a' and the common ratio be 'r'.

6th term of the G.P. is 46875
For the 6th term, n=6.
$a_n=ar^{n-1}$
Plug in n=6:
$a_6=ar^{6-1}$
Plug in $a_6=46875$:
$46875=ar^5$
$ar^5=46875$.......(i)


4th term of the G.P. is 375
For the 4th term, n=4.
$a_n=ar^{n-1}$
Plug in n=4:
$a_4=ar^{4-1}$
Plug in $a_4=375$:
$375=ar^3$
$ar^3=375$.......(ii)


To solve for r, divided equation (i) by (ii):
$\frac{ar^5}{ar^3}= \frac{46875}{375}$
$r^2=125$
square root both sides:
$r=\pm\sqrt{125} =\pm \sqrt{25*5}= \pm\sqrt{25}* \sqrt{5}$
$r=\pm5 \sqrt{5}$

Finding the 1st term 'a':
Case 1: When $r=-5 \sqrt{5}$
Plug in $r=-5 \sqrt{5}$ in equation (i):
$ar^3=375$
$a(-5 \sqrt{5})^3=375$
$-a*625 \sqrt{5} =375$
Divide both sides by $-625 \sqrt{5}$:
a=$- \frac{3}{5 \sqrt{5} }$

Case 2: When $r=5 \sqrt{5}$
Plug in $r=5 \sqrt{5}$ in equation (i):
$ar^3=375$
$a(5 \sqrt{5})^3=375$
$a*625 \sqrt{5} =375$
Divide both sides by $625 \sqrt{5}$:
a=$\frac{3}{5 \sqrt{5} }$


Finding the value of of the 9th term:
Case 1:
$a=- \frac{3}{5 \sqrt{5} }\text{, } r=-5 \sqrt{5} \text{ and }n=9\text{ (since 9th term n=9)}$
Plug in the above values in the formula:
$a_n=ar^{n-1}$
[tex]a_9=(- \frac{3}{5 \sqrt{5} })*(-5 \sqrt{5})^{9-1} \\ a_9=- \frac{3}{5 \sqrt{5} }*(5 \sqrt{5})^8 \\ a_9=-3(5 \sqrt{5})^7 \\ a_9=-29296875 \sqrt{5} [/tex]


Case:2
$a= \frac{3}{5 \sqrt{5} }\text{, } r=5 \sqrt{5} \text{ and }n=9\text{ (since 9th term n=9)}$
Plug in the above values in the formula:
$a_n=ar^{n-1}$
[tex]a_9=( \frac{3}{5 \sqrt{5} })*(5 \sqrt{5})^{9-1} \\ a_9= \frac{3}{5 \sqrt{5} }*(5 \sqrt{5})^8 \\ a_9=3(5 \sqrt{5})^7 \\ a_9=29296875 \sqrt{5} [/tex]


Answer:
$\text{ 9th term is :} -29296875 \sqrt{5} \text{ or }29296875 \sqrt{5}$