Give me a 3 example problems in geometric sequence and solve it
Answer (1)
1. The first term is 5, and the common ratio is 0.5 Find the 8th term.[tex]\mathsf{a_n = a_1 \cdot r^{n-1}}[/tex][tex]\mathsf{a_8 = 6 \cdot (0.5)^{7} = 6 \cdot \frac{1}{128} = \frac{6}{128} = \frac{3}{64}}[/tex]Answer: 3/642. Find two numbers between 2 and 16 so that the sequence forms a geometric sequence.[tex]\mathsf{2,\ x,\ y,\ 16}[/tex][tex]\mathsf{x = 2 \cdot r,\quad y = 2 \cdot r^2,\quad 16 = 2 \cdot r^3}[/tex][tex]\mathsf{r^3 = \frac{16}{2} = 8 \Rightarrow r = 2}[/tex]Thenx = 2 • 2 = 4y = 2 • 2² = 8Answers: 4 & 8The geometric sequence starts with a_1 = 10 , common ratio r = -2 . Find the sum of the first 5 terms.[tex]\mathsf{S_n = a_1 \cdot \frac{r^n - 1}{r - 1}}[/tex][tex]\mathsf{S_5 = 10 \cdot \frac{(-2)^5 - 1}{-2 - 1} = 10 \cdot \frac{-32 - 1}{-3} = 10 \cdot \frac{-33}{-3} = 10 \cdot 11 = 110}[/tex]Answer: 110
