Given: An infinite geometric series whose first term is 24 and a common ration of (-1/4), What is the exact..?

Question

sum of this infinte geometric series?

Answer 1

96/5 or 19.2

S = 24P where P= 1 -(1/4) + 1/16 - 1/64 + ... + (-1/4)^k+ ...

16P = 16 - 4 + 1 - (1/4) + 1/16 - 1/64 + ... = 12+P
15P = 12 or P = 4/5

so S = 24P = 96/5 = 19.2

in general,
S = a(1 + r + r^2 + r^3 + ...)
S(1-r) = a(1-r)(1 + r + r^2 + r^3 + ...)
S(1-r) = a[ (1 + r + r^2 + r^3 + ...) - (r + r^2 + r^3 + ...) ] = a* 1 = a

so S = a/(1-r)

Answer 2

For an infinite geometric series, the sum equals:

a / (1 - r) where "a" is the first term and "r" is the ratio.

Sum = 24 / (1 - (-1/4)) = 24 / (5/4)

=96/5

Answer 3

S=a/(1-r)
=24/[1-(-1/4)]
=24/[1+1/4]
=24/[5/4]
=24(4/5)
=96/5 or 19.2

Answer 4

19.2

You can use excel to verify this. Column A is the latest term (multiplying the previous term by -1/4) and column B is the
sum up to and including that term.


A B
2424.0000000000
-618.0000000000
1.519.5000000000
-0.37519.1250000000
0.0937519.2187500000
-0.023437519.1953125000
0.00585937519.2011718750
-0.00146484419.1997070313
0.00036621119.2000732422
-9.15527E-0519.1999816895
2.28882E-0519.2000045776
-5.72205E-0619.1999988556
1.43051E-0619.2000002861
-3.57628E-0719.1999999285
8.9407E-0819.2000000179
-2.23517E-0819.1999999955
5.58794E-0919.2000000011
-1.39698E-0919.1999999997
3.49246E-1019.2000000001
-8.73115E-1119.2000000000
2.18279E-1119.2000000000
-5.45697E-1219.2000000000
1.36424E-1219.2000000000
-3.41061E-1319.2000000000
8.52651E-1419.2000000000
-2.13163E-1419.2000000000
5.32907E-1519.2000000000
-1.33227E-1519.2000000000
3.33067E-1619.2000000000
-8.32667E-1719.2000000000
2.08167E-1719.2000000000
etc

Answer 5

Sum = a/(1-r),,,,,,, a is the first term, r is the common ratio

Sum= 24/(1+0.25)

= 19.2

Answer 6

S(infinity) = (a1)/(1 - r)

S(infinity) = (24)/(1 - (-1/4))
S(infinity) = 24/(1 + (1/4))
S(infinity) = 24/((4/4) + (1/4))
S(infinity) = 24/((4 + 1)/4)
S(infinity) = 24/(5/4)
S(infinity) = (24/1)/(5/4)
S(infinity) = (24/1)*(4/5)
S(infinity) = (96/5)
S(infinity) = 19.2

Answer 7

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