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

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 \(20.222\).