a recursive sequence

[george22]

a recursive sequence includes a (sub 1), a (sub 2), a (sub 3)....a(sub n) and a(sub 1) = 100. What is a (sub 50) when d = a (sub 1) + 0.2a (sub n - 1)?

 

[Gene]

a₁=100*1
a₂=100*(1+.2)
a₃=100*(1+.2+.2²)
a₄=100*(1+.2+.2²+.2³)
...

 

[soroban]

Hello, george22!

Some clarification, please . . .

\(\displaystyle \text{A recursive sequence includes: }\.a_1,\:a_2,\;a_3,\;.\,.\,.\,a_n\) and \(\displaystyle \,a_1\,=\,100\)

\(\displaystyle \text{What is }a_{50}\text{ when }d\:=\:a_1\,+\,0.2a_{n-1}\;?\)

Exactly what is \(\displaystyle d\) ?

It seems to stand for a "difference", but it is not constant.
\(\displaystyle \;\;\)It should have a subscript: \(\displaystyle \,d_1,\;d_2,\;d_3,\;.\,.\,.\)

As written: \(\displaystyle \,d\;=\;a_1\,+\,0.2a_{n-1}\), it says:
\(\displaystyle \;\;\)the difference is the first term plus 0.2 times the preceding term.


The first term is \(\displaystyle a_1\).

\(\displaystyle \;\;\)Then the difference is: \(\displaystyle \,d_1\;=\;a_1\,+\,0.2a_1\:=\:1.02a_1\)

The second term is: \(\displaystyle a_2\;=\;a_1\,+\,d_1\;=\;a_1\,+\,1.02a_1 \;=\;2.02a_1\)

\(\displaystyle \;\;\)Then the difference is: \(\displaystyle \,d_2\;=\;a_1\,+\,0.2(2.02a_1) \;=\;1.404a_1\)

The third term is: \(\displaystyle a_3\;=\;a_2\,+\,d_2\;=\;2.02a_1\,+\,1.404a_1\;=\;3.424a_1\)

. . . and we have a less attractive sequence (downright ugly!)


With: \(\displaystyle \,a_n\:=\:a_1\,+\,0.2a_{n-1}\) . . . Gene is correct.