Question

If 10term of gp is10and4th term is 4 andits 13th term is

Answer 1

The $13^{th}$ term of GP is $t_13$ = $25 \sqrt{\frac{2}{5}}$ .

• The $10^{th}$ term of the GP is 10.
• The $4^{th}$ term of the GP is 4.
• Now we have the formula for $n^{th}$ of the GP is given by

$t_n$ = $ar^{n-1}$

• We substitute the values in the above equation,

$t_4$ = $ar^{3}$

4 = $ar^{3}$    .....(Equation 1)

$t_10$ = $ar^{9}$

10 = $ar^{9}$    .....(Equation 2)

• Now dividing equation 2 by 1 we get

$r^{6}$ = $\frac{10}{4}$    .....(Equation 3)

${(r^{3})}^2$ = ${\frac{10}{4}}$

$r^{3}$ = $\sqrt{\frac{5}{2}}$   ....(Equation 4)

• Substituting the value in equation 1 we get

4 = a$\sqrt{\frac{5}{2}}$

a = 4 $\sqrt{\frac{2}{5}}$    ...(Equation 5)

• Now we need to find the value of $13^{th}$ term

$t_{13}$ = $ar^{12}$

$t_{13}$ = $a{(r^{6})}^2$

• We know that $r^{6}$ = $\frac{10}{4}$  (From equation 3) and a = 4 $\sqrt{\frac{2}{5}}$ (From equation 5)

$t_{13}$ = $4 \sqrt{\frac{2}{5}}{(\frac{10}{4})}^2$

$t_{13}$ = $25 \sqrt{\frac{2}{5}}$