Question

Find S9, the sum of the geometric series to the 9th term, in which a3=3.645 and a8=15

Answer 1

Answer:

The sum of the first nine terms of the sequence is 74.44.

Step-by-step explanation:

Geometric sequence concepts:

The nth term of a geometric sequence is given by the following equation.

$a_{n+1} = ra_{n}$

In which r is the common ratio.

This can be expanded for the nth term in the following way:

$a_{n} = a_{1}r^{n-1}$

In which $a_{1}$ is the first term.

Or even:

$a_{n} = a_{m}r^{n-m}$

The sum of the first n terms of a geometric sequence is given by:

$S_{n} = \frac{a_{1}(1 - r^{n})}{1 - r}$

Finding the common ratio:

$a_{3} = 3.645, a_{8} = 15$

$a_{n} = a_{m}r^{n-m}$

$a_{8} = a_{3}r^{8-3}$

$a_{3}r^{5} = a_{8}$

$3.645r^{5} = 15$

$r^{5} = \frac{15}{3.645}$

$r = \sqrt[5]{\frac{15}{3.645}}$

$r = 1.327$

Finding the first term:

$a_{3} = a_{1}r^{2}$

$a_{1} = \frac{a_{3}}{r^{2}}$

$a_{1} = \frac{3.645}{(1.327)^{2}}$

$a_{1} = 2.07$

Sum of the first nine terms:

$S_{9} = \frac{2.07*(1 - (1.327)^{9})}{1 - 1.327} = 74.44$

The sum of the first nine terms of the sequence is 74.44.

Answer 2

Answer:

S₉ = 9.84

Step-by-step explanation:

In a geometric series, the general formula for nth term is:

an = a₁rⁿ⁻¹

where,

an = nth term

a₁ = first term

r = common ratio

From this formula, we have:

a₃ = a₁r² = 3.645   --------- equation (1)

a₈ = a₁r⁷ = 15    ----------- equation (2)

dividing both of these equations, we get:

a₁r⁷/a₁r² = 15/3.645

r⁵ = 4.115

r = (4.115)^1/5

r = 1.33

Now, put this value in equation (1), we get:

a₁(1.33)² = 3.645

a₁ = 0.27

Now, the formula for the sum of geometric series upto nth term is:

Sn = a₁(rⁿ - 1)/(r - 1)

therefore,

S₉ = (0.27)(1.33⁹ - 1)/(1.33 - 1)

S₉ = 9.84