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
