Question

please explain #urgent
the sum to infinity of a convergent series is 243. the sum of the first five terms is 242. how do you determine the values of the common ratio and the first term

Answer 1

Use the formulas for both "sum of terms" and "sum of infinite number of terms"

Sum to infinity = a/(1-r)
a/(1-r) = 243 , ---- #1
( I recognized 243 as 3^5)
a = 243 - 243r

sum(5) = a(1 - r^5)/(1-r)
242 = a(1 - r^5)/(1-r) ---- #2

divide #2 by #1
1-r^5 = 242/243
r^5 =1 - 242/243
r^5 = 1/243
r^5 = 1/3^5
r = 1/3

back in #1
a = 243-243(1/3)
= 162

<b>a = 162, r = 1/3</b>

check:
terms are:
162 54 18 6 2 2/3 ...

sum(5) = 242
sum(of all) = 162/(1-1/3)
= 162/(2/3) = 243