how to solve what is the sum of the first ten terms of the geometric sequence 4,8,16,.....?
how to solve what is the sum of the first ten terms of the geometric sequence 4,8,16,.....?
\(\begin{array}{rcl} a_1 &=& 2^2 \\ a_2 &=& 2^3 \\ a_3 &=& 2^4 \\ \cdots \\ a_n &=& 2^{n+1} \end{array}\)
\(\begin{array}{rcrcl} s_n &=& a_1 &+& a_2+a_3+a_4+ \cdots + a_{n-1} + a_n \\ \hline s_n &=& 2^2 &+& 2^3+2^4+2^5+ \cdots + 2^n + 2^{n+1} \\ 2\cdot s_n &=& && 2^3+2^4+2^5+ \cdots + 2^n+2^{n+1} + 2^{n+2} \\ \hline s_n - 2\cdot s_n &=& 2^2 - 2^{n+2} \\ s_n(1 - 2) &=& 2^2 - 2^{n}\cdot 2^2 \\ -s_n &=& 2^2 - 2^{n}\cdot 2^2 \\ s_n &=& 2^{n}\cdot 2^2 - 2^2\\ s_n &=& 2^2\cdot (2^{n}-1)\\\\ \mathbf{s_n} &\mathbf{=}& \mathbf{4\cdot (2^{n}-1)} \\\\ s_{10} &=& 4\cdot (2^{10}-1)\\ s_{10} &=& 4\cdot (1024-1)\\ s_{10} &=& 4\cdot 1023\\ \mathbf{s_{10}} &\mathbf{=} & \mathbf{4092} \end{array}\)
The sum of the first ten terms of the geometric sequence is 4092
how to solve what is the sum of the first ten terms of the geometric sequence 4,8,16,.....?
\(\begin{array}{rcl} a_1 &=& 2^2 \\ a_2 &=& 2^3 \\ a_3 &=& 2^4 \\ \cdots \\ a_n &=& 2^{n+1} \end{array}\)
\(\begin{array}{rcrcl} s_n &=& a_1 &+& a_2+a_3+a_4+ \cdots + a_{n-1} + a_n \\ \hline s_n &=& 2^2 &+& 2^3+2^4+2^5+ \cdots + 2^n + 2^{n+1} \\ 2\cdot s_n &=& && 2^3+2^4+2^5+ \cdots + 2^n+2^{n+1} + 2^{n+2} \\ \hline s_n - 2\cdot s_n &=& 2^2 - 2^{n+2} \\ s_n(1 - 2) &=& 2^2 - 2^{n}\cdot 2^2 \\ -s_n &=& 2^2 - 2^{n}\cdot 2^2 \\ s_n &=& 2^{n}\cdot 2^2 - 2^2\\ s_n &=& 2^2\cdot (2^{n}-1)\\\\ \mathbf{s_n} &\mathbf{=}& \mathbf{4\cdot (2^{n}-1)} \\\\ s_{10} &=& 4\cdot (2^{10}-1)\\ s_{10} &=& 4\cdot (1024-1)\\ s_{10} &=& 4\cdot 1023\\ \mathbf{s_{10}} &\mathbf{=} & \mathbf{4092} \end{array}\)
The sum of the first ten terms of the geometric sequence is 4092