Expressions can be represented using summations
The series from least to greatest is: [tex]\sum\limits^6_{k = 0}27(\frac 13)^k[/tex], [tex]\sum\limits^5_{k = 0}50(\frac 18)^k[/tex] and [tex]\sum\limits^6_{k = 0}33(\frac 12)^k[/tex]
How to order the expressions
Start by evaluating the summation expressions
[tex]\sum\limits^6_{k = 0}27(\frac 13)^k[/tex]
Expand the above expression
[tex]\sum\limits^6_{k = 0}27(\frac 13)^k = 27 * (\frac 13)^0 + 27 * (\frac 13)^1 + 27 * (\frac 13)^2 + 27 * (\frac 13)^3 + 27 * (\frac 13)^4 + 27 * (\frac 13)^5 + 27 * (\frac 13)^6[/tex]
So, we have:
[tex]\sum\limits^6_{k = 0}27(\frac 13)^k = 27 * 1 + 27 * \frac 13 + 27 * \frac 19 + 27 * \frac 1{27} + 27 * \frac 1{81} + 27 * \frac 1{243} + 27 * \frac 1{729}[/tex]
Evaluate the sum
[tex]\sum\limits^6_{k = 0}27(\frac 13)^k = 27 +9 + 3 + 1 + \frac 1{3} + \frac 1{9} + \frac 1{27}[/tex]
Evaluate the sum
[tex]\sum\limits^6_{k = 0}27(\frac 13)^k = 40\frac {13}{27}[/tex]
[tex]\sum\limits^6_{k = 0}27(\frac 13)^k \approx 40.4815[/tex]
Also, we have:
[tex]\sum\limits^5_{k = 0}50(\frac 18)^k[/tex]
Expand the above expression
[tex]\sum\limits^5_{k = 0}50(\frac 18)^k = 50(\frac 18)^0 + 50(\frac 18)^1 +50(\frac 18)^2+50(\frac 18)^3 +50(\frac 18)^4 +50(\frac 18)^5[/tex]
Evaluate the exponents
[tex]\sum\limits^5_{k = 0}50(\frac 18)^k = 50 + 50(\frac 18) +50(\frac 1{64})+50(\frac 1{512}) +50(\frac 1{4096}) +50(\frac 1{32768})[/tex]
Evaluate
[tex]\sum\limits^5_{k = 0}50(\frac 18)^k \approx 57.143[/tex]
Also, we have the following expression
[tex]\sum\limits^6_{k = 0}33(\frac 12)^k[/tex]
Expand the above expression
[tex]\sum\limits^6_{k = 0}33(\frac 12)^k = 33(\frac 12)^0 +33(\frac 12)^1 +33(\frac 12)^2 +33(\frac 12)^3 +33(\frac 12)^4 +33(\frac 12)^5 +33(\frac 12)^6[/tex]
Evaluate the expression
[tex]\sum\limits^6_{k = 0}33(\frac 12)^k = 65.484375[/tex]
Rewrite the three expressions
[tex]\sum\limits^6_{k = 0}27(\frac 13)^k \approx 40.4815[/tex]
[tex]\sum\limits^5_{k = 0}50(\frac 18)^k \approx 57.143[/tex]
[tex]\sum\limits^6_{k = 0}33(\frac 12)^k = 65.484375[/tex]
So, the series from least to greatest is:
[tex]\sum\limits^6_{k = 0}27(\frac 13)^k[/tex], [tex]\sum\limits^5_{k = 0}50(\frac 18)^k[/tex] and [tex]\sum\limits^6_{k = 0}33(\frac 12)^k[/tex]
Read more about summation at:
https://brainly.com/question/6561461
