The population of a local species of mosquitos can be found using an infinite geometric series where a1 = 740 and the common ratio is one sixth. Write the sum in sigma notation, and calculate the sum (if possible) that will be the upper limit of this population.

Question

The population of a local species of mosquitos can be found using an infinite geometric series where a1 = 740 and the common ratio is  one sixth. Write the sum in sigma notation, and calculate the sum (if possible) that will be the upper limit of this population.

anonymous · Answer

$$\sum_{i=1 }^{ ∞}740(\frac{ 1 }{ 6})$$ but is the power to 1/6 ^i or i =1 also is it divergent? or does it have a sum im confused on the rest

anonymous · Answer

@ganeshie8 @iambatman @cwrw238

ganeshie8 · Answer

are you asking if the exponent would be $\large i$ or  $\large i-1$ ?

anonymous · Answer

yeah

anonymous · Answer

no i = 1 though not -

ganeshie8 · Answer

if $i$ starts from $1$,
the first term must evaluate to 740 right ?

ganeshie8 · Answer

$$\large    \sum_{i=1 }^{ \infty}740\left(\frac{ 1 }{ 6}\right)^{i-1} $$

ganeshie8 · Answer

the exponent has to be $\large i-1$
otherwise you wont get the first term as 740

anonymous · Answer

oh yeah my bad sorry couldn't see the minus clearly.

anonymous · Answer

now how do I find if its divergent? thats what Im most confused on

ganeshie8 · Answer

Notice that the common ratio is 1/6 which is less than 1
so the geometric series converges

ganeshie8 · Answer

$$\large    \text{if } |r| < 1, \text{ then the series converges} $$

ganeshie8 · Answer

use the infinite geometric series formula to evaluate the sum

ganeshie8 · Answer

$$\large  \text{infinite sum} = \dfrac{a}{1-r}$$

anonymous · Answer

it is 888

anonymous · Answer

thank you

ganeshie8 · Answer

888 is $\large \color{red}{\checkmark}$
good job !!