Question

The second and fifth terms of a geometric sequence are 750 and -6 respectively. Find the common ratio and first term of the sequence.

Answer 1

Step-by-step explanation:

The second term =ar=750---------(I)

the fifth term =ar⁵=-6-----------(ii)

now from eqn i

a=750/r

from eqn ii

750/r*r⁵=750r⁴=-6

r⁴=-6/750

r⁴=-1/125

r⁴=1⁴/5⁴

r=1/5

from eqn i

ar=750

a*1/5=750

a=3750

Answer 2

Given:

Second term of a geometric sequence = 750

Fifth term of a geometric sequence = - 6

To find :

The common ratio and first term of the sequence.

Formula to be used:

$t_n = ar^{n-1}$

Solution:

Step 1 of 2:

From the given values,

$t_2 = 750$

$t_n = ar^{n-1}$

Therefore,

$t_2 = ar^{1}$

$ar^{1}$ = 750  -------------Eq(1)

$t_5 = -6$

$t_5 = ar^4$

$ar^{4}$ = -6     -------------Eq(2)

To find r, divide Eq(1) by Eq(2)

$\frac{ar^{4} }{a r^1} = \frac{-6}{750}$

$r^3 = -\frac{1}{125}$

r = - ∛$\frac{1}{125}$

r = $-\frac{1}{5}$

Step 2 of 2:

Substitute 'r' in Eq(1)

ar = 750

a $(-\frac{1}{5})$ = 750

a = 750 × $(-\frac{5}{1} )$

a = - 3750

Final answer:

The common ratio of the sequence, r =  $-\frac{1}{5}$

The first term of the sequence, a = - 3750