As n → +∞, Sn → +6
Sn approaches a horizontal asymptote when y=6. There is a y-intercept at (0,1).
tn =
As n → +∞, Sn → +7
There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive seven.
tn=
As n → +∞, Sn → +8
There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive eight. Sn approaches a horizontal asymptote when y=8. There is a y-intercept at (0,1).
Consider the following sequence where x=5 and a=2 when 0≤n≤9.
tn=
As n → +∞, Sn → +6
Let a=2 and calculate various positive values for x:
In the graph, when x is approaching infinite the Sn values are increasing steadily. When various values are used for x then there is an exponential growth.
Let a=3 then calculate for various positive values of x:
In the graph, when x is approaching infinite the Sn values are increasing steadily. When various values are used for x then there is an exponential growth.
Evidence:
tn=
As n → +∞, Sn → +6
Sn approaches a horizontal asymptote when y=6. There is a y-intercept at (0,1).
tn =
As n → +∞, Sn → +7
There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive seven. Sn approaches a horizontal asymptote when y=7. There is a y-intercept at (0,1).
tn =
As n → +∞, Sn → +8
There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive eight. Sn approaches a horizontal asymptote when y=8. There is a y-intercept at (0,1).
tn =
As n → +∞, Sn → + 9
There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive nine. Sn approaches a horizontal asymptote when y=9. There is a y-intercept at (0,1).
tn =
As n → +∞, Sn → + 10
There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive ten. Sn approaches a horizontal asymptote when y=10. There is a y-intercept at (0,1).
tn=
As n → +∞, Sn → + 9
. Sn approaches a horizontal asymptote when y=9. There is a y-intercept at (0,1).
tn =
As n → +∞, Sn → + 11
There is a horizontal asymptote as n approaches positive infinite (∞).
tn=
As n → +∞, Sn → + 15
There is a horizontal asymptote as n approaches positive infinite (∞). As n approaches positive infinite then Sn will approach positive fifteen. Sn approaches a horizontal asymptote when y=15. There is a y-intercept at (0,1).
tn=
As n → +∞, Sn → + 17
There is a horizontal asymptote as n approaches positive infinite (∞). Sn approaches a horizontal asymptote when y=17. There is a y-intercept at (0,1).
Checks with various numbers:
tn =
tn =
tn =
Conclusion:
The limitation to this general statement is that the scope of the evidence is limited to only a few combinations of x and a .
The continuous observation of various graphs with different values used in place of a and x provide evidence for the general statement.
