Answer :
Let's tackle each sequence separately to find the rule and determine the 50th term.
Part a: Sequence 1, 10, 20, 40, 180
Finding the Rule:
The sequence seems irregular and might require a close inspection. Let's try finding a pattern by examining ratios between consecutive terms.
The terms are 1, 10, 20, 40, 180.
- From 1 to 10, the factor is 10.
- From 10 to 20, the factor is 2.
- From 20 to 40, the factor is 2.
- From 40 to 180, the factor is 4.5.
This doesn't seem to exhibit a straightforward arithmetic or geometric sequence.
Given the inconsistent pattern, it's possible that a unique mathematical function defines this sequence rather than a simple arithmetic or geometric pattern.
Finding the 50th Term:
Without a clear arithmetic or geometric rule evident and no apparent polynomial pattern, determining the 50th term directly is challenging without additional context or rules.
As such, further context or rule information is needed to proceed accurately.
Part b: Sequence 25, 125, 625, 3125
- Finding the Rule:
This is a geometric sequence. Let's find the common ratio.
- Take two consecutive terms: 125/25 = 5 and 625/125 = 5 and 3125/625 = 5.
Therefore, the common ratio [tex]r[/tex] is 5.
The nth term of a geometric sequence can be found using the formula:
[
a_n = a_1 , r^{(n-1)}]
- Here, [tex]a_1 = 25[/tex] and [tex]r = 5[/tex].
- Finding the 50th Term:
Apply the formula for the 50th term:
[tex]a_{50} = 25 \, (5^{49})[/tex]Evaluating this expression will give us the 50th term.
- Note: Calculating [tex]5^{49}[/tex] will give an enormous number, so use appropriate computational tools to evaluate it accurately.
