Answer :

Let's tackle each sequence separately to find the rule and determine the 50th term.

Part a: Sequence 1, 10, 20, 40, 180

  1. 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.

  2. 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

  1. 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].

  1. 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.

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