What is the 8th term of the geometric sequence if #a_3 = 108# and #a_5 = 972#?

Answer 1

#26244#

In a geometric sequence is valid the following rule

#a_(i+1)=k*a_i#, where #i in NN#

So

#a_5=k*a_4=k*(k*a_3)# #a_5=k^2*a_3# #972=k^2*108# => #k^2=9# => #k=3#

By the same token

#a_8=k^(8-5)*a_5# #a_8=k^3*a_5=3^3*972=27*972# => #a_8=26244#
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Answer 2

#T_8 = ar^7 = 12 xx 3^7 = 26 244#

Before we can find an unknown term of a geometric sequence, we need to know the first term #(a)# and the common ratio #(r)#
Each term can be written as #T_n = ar^(n-1)#

Let's divide the two terms we have been given, their formulae and their values:

#(T_5)/(T_3) = (ar^4)/(ar^2) = 972/108#
The following happens: #(cancelar^4)/(cancelar^2) = 972/108#
Subtract the indices: #a^2 = 9 " " rArr r = 3#
In #T_3 = ar^2, if r = 3,# then #a(3^2) = 108#
#9a = 108 " "rArr a = 12#
Great! now we have #a and r# so we can find any term we want.
#T_8 = ar^7 = 12 xx 3^7 = 26 244#
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Answer 3

To find the 8th term of the geometric sequence, we need to first find the common ratio (r) of the sequence using the given terms (a_3) and (a_5).

Given (a_3 = 108) and (a_5 = 972),

We have:

(a_3 = a_1 \times r^2 = 108)

(a_5 = a_1 \times r^4 = 972)

Dividing the second equation by the first equation:

(\frac{a_5}{a_3} = \frac{a_1 \times r^4}{a_1 \times r^2} = r^2 = \frac{972}{108} = 9)

So, (r = \sqrt{9} = 3)

Now that we have the common ratio, we can find the first term ((a_1)) using the third term ((a_3)):

(a_3 = a_1 \times r^2 = 108)

(a_1 = \frac{a_3}{r^2} = \frac{108}{3^2} = 12)

Now, to find the 8th term ((a_8)), we use the formula for the nth term of a geometric sequence:

(a_n = a_1 \times r^{(n-1)})

(a_8 = a_1 \times r^{(8-1)})

(a_8 = 12 \times 3^{(8-1)})

(a_8 = 12 \times 3^7)

(a_8 = 12 \times 2187)

(a_8 = 26244)

So, the 8th term of the geometric sequence is 26244.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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