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
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
Answers
Reiny
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
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