Question

∑ ∞ n = 1 − 4 ( 1 3 ) n − 1

Answer 1

Answer:

6

Step-by-step explanation:

The correct notation is: Summation from n = 1 to infinity of the expression:

$4(\frac{1}{3} )^{n-1}$

If we expand this notation, we will find that the given summation represent the terms of a G.P and we have to find the sum of infinite terms of a G.P. By placing n=1,2,3,4,... we can find the first few terms as shown below:

$4(\frac{1}{3} )^{1-1}=4\\\\ 4(\frac{1}{3} )^{2-1}=4(\frac{1}{3} ) \\\\ 4(\frac{1}{3} )^{3-1} =4(\frac{1}{3} )^{2}$

Its obvious from the above result, that the given summation represent a G.P with first term equal to 4 and common ratio of 1/3. The formula of infinite G.P is:

$S=\frac{a_{1} }{1-r}$

Using the value of a1 = 4 and r = 1/3 we can calculate the given summation

$S=\frac{4}{1-\frac{1}{3} }=6$

Therefore, the given summation is equal to 6.