Question

Find the 9th term of the geometric sequence 5, -25, 125, ...

Answer 1

Answer:  1,953,125

This is one single value and it is just a bit under 2 million.

Or more accurately, it's a bit over 1.9 million.

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Explanation:

• a = 5 = first term
• r = -5 = common ratio

Note that dividing any term by its previous term gets us the common ratio

• r = term2/term1 = -25/5 = -5
• r = term3/term2 = 125/(-5) = -5

The r value must stay the same the entire time, or else the sequence isn't geometric.

The nth term of any geometric sequence is a*(r)^(n-1). With the 'a' and 'r' values we found, we update that to 5(-5)^(n-1)

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To verify that is the correct nth term expression, plug in various values of n to compare it with the given sequence.

If we tried n = 2 for instance, then we find the 2nd term is

5(-5)^(n-1) = 5(-5)^(2-1) = -25

which matches what your teacher gave you. I'll let you verify the other terms.

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The last thing we need to do is plug in n = 9 and simplify

5(-5)^(n-1)

5(-5)^(9-1)

5(-5)^8

5(390625)

1,953,125  this is one single value (rather than 3 separate values)