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March 29, 2024, 03:51:59 pm

Author Topic: Geometric Sequence Question  (Read 1659 times)  Share 

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BakerDad12

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Geometric Sequence Question
« on: November 04, 2019, 07:33:46 am »
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Hi all, I have two  questions on series and sequences I am unable to complete. Q1). Which term of the sequence 8, -4, 2,.., is 1/128?

Q2). Find the value of n if the nth term of the sequence is -2, 3/2, -9/8,..., is -81/128?

When I attempt these I get negative logs, and I don't know how to handle them.

Thanks for your help guys!

fun_jirachi

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Re: Geometric Sequence Question
« Reply #1 on: November 04, 2019, 03:31:46 pm »
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Hi there!

Recall that the formula for the nth term of a geometric series is \(T_n = ar^{n-1}\).

For the first one, we have that

And hence n = 11.

Similarly for the second one, we have that

And hence n=5.

The key here is to note what sign the term is. In each case, since the common ratio is negative, if the term is positive, the power must be even, and if the term is negative, the power must be odd. In this way, we can get rid of the minus signs and perform logarithms normally; or if the numbers are recognisable enough we can just put in the powers. In both cases above, each of the numbers was positive, and hence we could just remove the negative without consequence as -1 to any even power is just 1.

Hope this helps :)
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BakerDad12

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Re: Geometric Sequence Question
« Reply #2 on: November 10, 2019, 11:04:45 am »
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Hi, thanks so much for the great reply! It did help, except it left me with one question. "In each case, since the common ratio is negative, if the term is positive, the power must be even, and if the term is negative, the power must be odd. In this way, we can get rid of the minus signs and perform logarithms normally"

How can we get rid of the minus sign just because of this?

fun_jirachi

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Re: Geometric Sequence Question
« Reply #3 on: November 11, 2019, 03:44:38 pm »
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Sorry for not clarifying properly!

Basically what I'm saying is that if the common ratio is negative, the power essentially determines the parity of the term ie. whether it is odd or even. Say we had a geometric sequence with first term 1, and common ratio -0.5 - it'd go something like this: 1, -0.5, 0.25, -0.125, 0.0625... and so on. Note that for every odd term, the power on top of the common ratio in the formula arn-1 is odd: for the second term, -0.5, we have (1)(-0.5)2-1.

Note also, that \((ab)^n=a^n \times b^n\); this means that essentially, if we have any negative number to an odd power, we can split them up to form -1 to that same odd number, and the absolute value of that number to that same power. ie. (-a)n = (-1)n x an.

Remembering that -1 to any odd power is just -1, and combining this with the first paragraph, we essentially just stick a minus out the front and leave the common ratio positive for odd powers ie. negative terms - ie. we can just remove them all together, since both sides are negative. Then you can just perform a logarithm as usual.

It's a similar case for a positive number/even power, because we have -1 to an even power, and that is just 1; there are in actuality no negative signs to even contend with, so you can just do a log anyway.

Hope this helps :)
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