Question

A1=-3, r=4, Sn=-4095

Answer 1

The Solution to Number 25

To find the number of terms "n", recall that

`\begin{gathered} \text{Sum of a G.P is given as } \\ S_n\text{ }\equiv\text{ }(a(1-r^n))/(1-r)\text{ if r is less than 1 or }S_{n_{}}\text{ }\equiv\text{ }(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{ = }(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}`
`\begin{gathered} \text{From }S_n\text{ = }(a(1-r^n))/(1-r)\text{ substituting each values we have;} \\ 52428_{}\text{ = }(-4(1-(-4)^n))/(1-(-4))\text{ } \\ 52428_{}\text{ = }(-4+4(-4)^n)/(1+4)\text{ } \\ 52428_{}\text{ = }(-4+4(-4)^n)/(5)\text{ } \\ 52428\text{ }*\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}`

Therefore, the number of terms n for question 25 is given as

n = 8 terms.