16,_,_,_ rule:3. 1,4,16,_,_,_ rule: 4. 3,9,27,_,_,

Questions

August 2022 1 8 Report

Learning task 2: Find the next three terms in each sequence then write the rule for finding the nth term. Write your answer on a separate sheet of paper.

1. 2,6,18,_,_,_
rule:

2. 1/2,1/4,1/8,1/16,_,_,_
rule:

3. 1,4,16,_,_,_
rule:

4. 3,9,27,_,_,_
rule:

help me :')

Answers & Comments

MrFahrenheit

Verified answer

ARITHMETIC & GEOMETRIC SEQUENCE

Arithmetic sequence is a kind of sequence by which the two consecutive terms have a difference and it is constant throughout all terms of the sequence so it is known as the common difference. This also means that each term can be obtain by adding or subtracting the common difference to the existing terms to obtain the next following or preceding terms. When we want to solve for the nth term of an arithmetic sequence, the formula that could be used is:

$a_n = a_1 + (n-1)d$

where:

$a_n$ refers to the nth term of the given arithmetic sequence
$a_1$ refers to the first term of the given arithmetic sequence
n refers to the order of the term in the sequence
d refers to the common difference

The common difference of the arithmetic sequence is positive if the terms in the sequence is getting larger in value as it progresses, but its common difference is negative if the terms in the sequence is getting lesser in value as it progresses.

The general formula for obtaining the nth term of the arithmetic sequence can also be algebraically manipulated if we are calculating for other unknowns, other than the nth term.

On the other hand, geometric sequence is a kind of sequence by which the two consecutive terms have a quotient which is constant in all terms included in the said sequence, so it is known as the common ratio. This also means that each term can be obtain by multiplying the common ratio to each term to obtain the next following or preceding terms.

When we want to know the nth term of the geometric sequence, the formula that can be used is:

$a_n = a_{1}r^{n-1}$

where:

$a_n$ refers to the nth term of the geometric sequence
$a_1$ refers to the first term of the geometric sequence
r refers to the common ratio
n refers to the order of the nth term in the sequence

The common ratio is an integer if the succeeding terms in the given geometric sequence are getting larger in value. However, the common ratio may be a rational number (fraction form) if the succeeding terms in the given geometric sequence are getting smaller in value.

Its general formula can also be manipulated when we are looking for other unknown variables.

In this activity, we will deal with geometric sequence. Our task is to determine the next three terms and the rule for finding the nth term. Given this first set of sequence:

2, 6, 18, ...

The first step that we need to do is to pick a pair of consecutive terms. In this case, we can pick 2 and 6. Then, let us divide the second term by the first term. We should not worry about the quotient of other pair of consecutive terms since this is a geometric sequence which means the quotient or ratio in all pairs are common or the same. Going back to division: 6 ÷ 2 = 3. So 3 is the common ratio. We will multiply this to the last term in the sequence which is 18 in order to get the next 3 terms. We repeat the process as we go on each term.

18 x 3 = 54
54 x 3 = 162
162 x 3 = 486

Therefore, the next three terms of this geometric sequence is 54, 162 and 486.

In order to obtain the rule, we can manipulate the geometric sequence formula mentioned above, where $a_1 = 2$ and $r = 3$ . So:

$>So this is the general rule in finding the nth term of the given geometric sequence.Let us try this another sequence:1/2, 1/4, 1/8, 1/16, ...Consecutive Pairs: 1/2 and 1/4Divide: 1/4 ÷ 1/2 = 1/4 x 2 = 2/4 = 1/2Thus, the common ratio is 1/2.For the general rule:<img src=$