Question

write the sum using summation notation, assuming the suggested pattern continues. 3 - 15 + 75 - 375 + ...

Answer 1

Answer:

Sn = ∑ 3(-5)^n, from n = 0 to n = n

Step-by-step explanation:

* Lets study the geometric pattern

- There is a constant ratio between each two consecutive numbers

- Ex:

# 5  ,  10  ,  20  ,  40  ,  80  ,  ………………………. (×2)

# 5000  ,  1000  ,  200  ,  40  ,  …………………………(÷5)  

* General term (nth term) of a Geometric pattern

- U1 = a  ,  U2  = ar  ,  U3  = ar^2  ,  U4 = ar^3  ,  U5 = ar^4

- Un = ar^n-1, where a is the first term , r is the constant ratio between

 each two consecutive terms, n is the position of the number

# Ex: U5 = ar^4  ,  U7 = ar^6  ,  U10 = ar^9  ,  U12 = ar^11

- The sum of first n terms of a Geometric Pattern is calculate from

 Sn = a(1 - r^n)/(1 - r)

- The summation notation is ∑ a r^n, from n = 0 to n = n

* Now lets solve the problem

∵ The terms if the sequence are:

  3 , -15 , 75 , -375 , ........

∴ a = 3

∵ r = ar/a

∵ ar = -15 and a = 3

∴ r = -15/3 = -5

∵ Sn = a(1 - r^n)/(1 - r)

∴ Sn = 3[1 - (-5)^n]/[1 - (-5)] = 3[1 - (-5)^n]/6 = 1/2[1 - (-5)^n]

- By using summation notation

∵ Sn = ∑ a r^n , from n = 0 to n = n

∴ Sn = ∑ 3(-5)^n