Question 6 For the series $$ 27,-9,3,-1 \ldots $$, find: (i) the first term and the common ratio (ii) the sum of the first 6 terms

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To find the first term and the common ratio of the given series, we can use the formula for the nth term of a geometric sequence:

$$ a_n = a_1 \times r^{(n-1)} $$

where:
- $$ a_n $$ is the nth term of the sequence
- $$ a_1 $$ is the first term of the sequence
- $$ r $$ is the common ratio
- $$ n $$ is the term number

Given series: $$ 27, -9, 3, -1 \ldots $$

(i) Finding the first term and the common ratio:
- The first term $$ a_1 $$ is 27.
- The common ratio $$ r $$ can be found by dividing any term by the previous term. Let's divide the second term by the first term:
$$ r = \frac{-9}{27} = -\frac{1}{3} $$

(ii) Finding the sum of the first 6 terms:
- The sum of the first n terms of a geometric sequence can be found using the formula:
$$ S_n = \frac{a_1(1 - r^n)}{1 - r} $$

Substitute the values:
- $$ a_1 = 27 $$
- $$ r = -\frac{1}{3} $$
- $$ n = 6 $$

$$ S_6 = \frac{27(1 - (-\frac{1}{3})^6)}{1 - (-\frac{1}{3})} $$

Now, let's calculate the sum of the first 6 terms.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{27\left(1-\left(\frac{-1}{3}\right)^{6}\right)}{\left(1-\left(\frac{-1}{3}\right)\right)}$$
- step1: Remove the parentheses:
  $$\frac{27\left(1-\left(\frac{-1}{3}\right)^{6}\right)}{1-\left(\frac{-1}{3}\right)}$$
- step2: Rewrite the fraction:
  $$\frac{27\left(1-\left(-\frac{1}{3}\right)^{6}\right)}{1-\left(\frac{-1}{3}\right)}$$
- step3: Subtract the numbers:
  $$\frac{27\times \frac{728}{729}}{1-\left(\frac{-1}{3}\right)}$$
- step4: Rewrite the fraction:
  $$\frac{27\times \frac{728}{729}}{1-\left(-\frac{1}{3}\right)}$$
- step5: Remove the parentheses:
  $$\frac{27\times \frac{728}{729}}{1+\frac{1}{3}}$$
- step6: Multiply the numbers:
  $$\frac{\frac{728}{27}}{1+\frac{1}{3}}$$
- step7: Add the numbers:
  $$\frac{\frac{728}{27}}{\frac{4}{3}}$$
- step8: Multiply by the reciprocal:
  $$\frac{728}{27}\times \frac{3}{4}$$
- step9: Reduce the numbers:
  $$\frac{182}{9}\times 1$$
- step10: Multiply:
  $$\frac{182}{9}$$
The sum of the first 6 terms of the given series is $$20\frac{2}{9}$$ or $$20.222$$.
First term: 27 Common ratio: -1/3 Sum of the first 6 terms: 20.222

Question 6
For the series , find:
(i) the first term and the common ratio
(ii) the sum of the first 6 terms

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Question 6 For the series , find: (i) the first term and the common ratio (ii) the sum of the first 6 terms