The sum of the first n terms of a series is 1-(3/4). Obtain an expression for the nth term of the series. Prove that the series is geometric, and state the values of the first term and the common ratio. Please show workings

The n'th term is ar^(n-1)

Sn = a(1-r^n)/(1-r) = 1/4
4a(1-r^n) = 1-r
4a - 4ar^n = 1-r
4a - 4ar^(n-1)*r = 1-r
ar^(n-1) = (1-r-4a)/(4r)

I can't see how you can solve for r and a. There are many possible solutions.

n=1: a=1/4

n=2: a(1+r) = 1/4
a = 1, r = -3/4
a = -1/3, r = -7/4

n=3: a(1+r+r^2) = 1/4
...
I think there must be something missing here. Or a typo.

To obtain an expression for the nth term of the series, let's first write the given sum in a general form. The sum of the first n terms of a series is typically represented as Sn. So, let's write the given sum as Sn:

Sn = 1 - (3/4)

Now, to find the expression for the nth term, we need to determine the difference between consecutive terms. Let's assume the first term of the series is a and the common ratio is r.

The formula for the sum of a finite geometric series is given by:

Sn = a(1 - r^n) / (1 - r) (1)

where 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.

Comparing this formula with our given sum, Sn = 1 - (3/4), we can see that the expression 1 - r^n in formula (1) corresponds to (3/4) in our given sum.

Therefore, we have:

1 - r^n = 3/4

Now, we need to prove that the series is geometric. In a geometric series, each term can be obtained by multiplying the previous term by a constant ratio.

From the equation 1 - r^n = 3/4, we can manipulate it to find the common ratio, 'r'. Rearranging the equation, we get:

1 = 3/4 + r^n

Simplifying further:

1 = (3 + 4r^n) / 4

Cross-multiplying:

4 = 3 + 4r^n

Rearranging the equation:

4r^n = 4 - 3

4r^n = 1

Dividing both sides by 4:

r^n = 1/4

Now, we have found the relationship r^n = 1/4, which shows that the series is indeed a geometric series. The value of the first term, 'a,' is 1, and the common ratio, 'r,' can be found by taking the nth root of 1/4.

Taking the square root of both sides:

(r^n)^(1/2) = (1/4)^(1/2)

r^(n/2) = 1/2

Since we are looking for the value of 'r,' we know that n/2 = 1.

Simplifying further:

r = (1/2)^(2/1) = 1/2

Therefore, the first term of the series, 'a,' is 1, and the common ratio, 'r,' is 1/2.