2 Answers

Ram Grandhi · Answer 1 · 2023-10-15T04:06:48+0000

Answer:

Number Sequence: a set of numbers that follow a pattern or a rule, where each number in the sequence is called a term.

Arithmetic Sequence: has a constant difference between each term, so the difference between each term is the same.

Geometric Sequence: has a constant ratio (multiplier) between each term, so each term is multiplied by the same number.

To determine the type of sequence, calculate the differences between the terms:

$7.5 \underset{-3.75}{\longrightarrow} 3.75 \underset{-1.875}{\longrightarrow} 1.875$

Therefore, this is not an arithmetic sequence, as the difference between the terms is not the same.

General form of a geometric sequence:

a_n=ar^(n-1)

(where a is the first term and r is the common ratio)

To find the common ratio r, divide consecutive terms:

$\implies r=(a_2)/(a_1)=(3.75)/(7.5)=0.5$

Therefore:

a_n=7.5(0.5)^(n-1)

Finding the 4th and 5th terms:

$\implies a_4=7.5(0.5)^(4-1)=0.9375$

$\implies a_5=7.5(0.5)^(5-1)=0.46875$

Part (a)

$\large \begin{array} c \cline{1-6} n & 1 & 2 & 3 & 4 & 5 \\\cline{1-6} t(n) & 7.5 & 3.75 & 1.875 & 0.9375 & 0.46875 \\\cline{1-6}\end{array}$

Part (b)

Geometric sequence

Part (c)

a_n=7.5(0.5)^(n-1)

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