Respuesta :
The Solution to Number 25
To find the number of terms "n", recall that
[tex]\begin{gathered} \text{Sum of a G.P is given as } \\ S_n\text{ }\equiv\text{ }\frac{a(1-r^n)}{1-r}\text{ if r is less than 1 or }S_{n_{}}\text{ }\equiv\text{ }\frac{a(r^n-1)}{r-1}\text{ if r is greater than 1. } \\ So\text{ since the common ratio r = -4 which is less than 1, then we will use} \\ S_n\text{ = }\frac{a(1-r^n)}{1-r}\text{ , where r= -4, a = first term = }-4\text{ } \\ \text{and }S_n\text{ }=\text{ sum of terms of the G.P = }52428 \end{gathered}[/tex][tex]\begin{gathered} \text{From }S_n\text{ = }\frac{a(1-r^n)}{1-r}\text{ substituting each values we have;} \\ 52428_{}\text{ = }\frac{-4(1-(-4)^n)}{1-(-4)}\text{ } \\ 52428_{}\text{ = }\frac{-4+4(-4)^n}{1+4}\text{ } \\ 52428_{}\text{ = }\frac{-4+4(-4)^n}{5}\text{ } \\ 52428\text{ }\times\text{ 5 = }-4+4(-4)^n \\ 262140\text{ + 4 }=\text{ 4}(-4)^n \\ \text{ 4}(-4)^n\text{ = 262144 } \\ \text{Dividing both sides of the equation by 4 we have;} \\ \frac{\text{ 4}(-4)^n}{4}\text{ = }\frac{\text{262144 }}{4} \\ \\ \text{ }(-4)^n\text{ = 65536} \\ (-4)^n\text{ = }(-4)^8\text{ note that 65536 = }4^8\text{ or }(-4)^8 \\ \text{Cancelling out the bases (-4) we have} \\ n\text{ = 8} \end{gathered}[/tex]Therefore, the number of terms n for question 25 is given as
n = 8 terms.
