For the series 1/2 , 1/4 , 1/6 , 1/8 , 1/10, 1/12, 1/14, 1/16 what is

the nth term?

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The given terms of the series are:

1/2, 1/4, 1/6, 1/8, 1/10 1/12, 1/14, 1/16 ...

Thus

Term number 1 = 1/2 = 1/(1*2)

Term number 2 = 1/4 1/(2*2)

Term number 3 = 1/6 = 1/(3*2)

Term number 4 = 1/8 = 1/(4*2)

Term number 5 = 1/10 = 1/(5*2)

Term number 6 = 1/12 = 1/(6*2)

Term number 7 = 1/14 = 1/(7*2)

Term number 8 = 1/16 = 1/(8*2)

Term number n = 1/(n*2)

Answer:

nth term = 1/(n*2)

The given series is not an AP as 1/2- 1/4 is not equal to 1/4- 1/6 , therefore the difference between the terms is not common. Also it is not a GP as (1/4) / (1/2) is not equal to (1/8)/ (1/6) , therefore they do not have a common ratio.

But if we see the terms carefully we notice that 2, 4, 6, 8...are terms of an AP. The series given here is called a harmonic series where the terms of the series are the reciprocal of the terms of an AP. So the nth term of the series is the reciprocal of 2+(n-1)*2 or 1/ [2+ (n-1)*2]

Therefore the nth term of the series is 1/ [2+ (n-1)*2]

The nth term can be calculated simply as (-1)^(n+1)/(2n). Where n starts at 1 for the first term and goes to infinity.

To find the nth term of

1/2 , 1/4 , 1/6 , 1/8 , 1/10, 1/12, 1/14, 1/16.

The numarator of all the terms is 1. So the numwrator of the nth term is also 1.

The denominator of the terms are 2, 4 , 8, 6 etc. Or the 1st term ha the denominator 1*2,

the 2nd term has the denominator 2*2.

The 3rd term has the denominator 3*2.

So the denominator of the nth term = n*2.

So the nth term of the sequence = 1/(2n).

The given series is not an AP as 1/2- 1/4 is not equal to 1/4- 1/6 , therefore the difference between the terms is not common. Also it is not a GP as (1/4) / (1/2) is not equal to (1/8)/ (1/6) , therefore they do not have a common ratio.

But if we see the terms carefully we notice that 2, 4, 6, 8...are terms of an AP. The series given here is called a harmonic series where the terms of the series are the reciprocal of the terms of an AP. So the nth term of the series is the reciprocal of 2+(n-1)*2 or 1/ [2+ (n-1)*2]

Therefore the nth term of the series is 1/ [2+ (n-1)*2]

This doesn't work as it gives, 1/2, 1/4, 1/6, ... No negative values are given.

As you can notice the denominator is increasing by 2, but another way to think of it is its increasing by the number *2 so n*2, but since its the denominator, the one remains the same 1/n*2 is the pattern.

From the pattern above you should look at the denominator. As you can see, 2 later changes to 4 and then 6 and so on. By looking at this you can see that the denominator adds 2 each time. So to get the nth term of this sequence you have to use 1/2n

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