Step 1
The series is \(2,−12,72,−432,2592,...\)
check the ratio
\(\frac{-12}{2} = -6\)

\(\frac{72}{-12} = -6\)

\(\frac{-432}{72} = -6\) this is geometrix series with common ratio r, where \(r = -6\) the sequence is geometric (b) the general term \(a_i = a_1 r^(i-1)\) where \(a_1\) is the first term , r is the common ratio. substitute the values \(a_i = 2(−6)^(i−1)\) hence, the expression for \(a_i is a_i = 2(−6)^{i -1}\) Step 2 the sum of first 10 terms is given by the formula \(\frac{a(1-r^n)}{1- r}\) substitute the values to get the sum of first 10 terms \(S_n = \frac{a(1-r^n)}{1- r}\)

\(= \frac{2(1-(-6)^n)}{1-(-6)}\)

\(= {2(1-(-6)^n)}{1+6}\)

\(= 17276050\) hence, the sum of first 10 terms is given by 17276050

\(\frac{72}{-12} = -6\)

\(\frac{-432}{72} = -6\) this is geometrix series with common ratio r, where \(r = -6\) the sequence is geometric (b) the general term \(a_i = a_1 r^(i-1)\) where \(a_1\) is the first term , r is the common ratio. substitute the values \(a_i = 2(−6)^(i−1)\) hence, the expression for \(a_i is a_i = 2(−6)^{i -1}\) Step 2 the sum of first 10 terms is given by the formula \(\frac{a(1-r^n)}{1- r}\) substitute the values to get the sum of first 10 terms \(S_n = \frac{a(1-r^n)}{1- r}\)

\(= \frac{2(1-(-6)^n)}{1-(-6)}\)

\(= {2(1-(-6)^n)}{1+6}\)

\(= 17276050\) hence, the sum of first 10 terms is given by 17276050