Post by: BakerDad12 on November 04, 2019, 07:33:46 am
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!
Post by: fun_jirachi on November 04, 2019, 03:31:46 pm
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 :)
Post by: BakerDad12 on November 10, 2019, 11:04:45 am
How can we get rid of the minus sign just because of this?
Post by: fun_jirachi on November 11, 2019, 03:44:38 pm
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 :)