For the Sequence 2,6,10,14. Write the recursive rule, and a explicit rule

It is observed that the difference between the consecutive terms remains constant) with common difference d = 4
We know that if the difference between the consecutive terms remains constant), then the series is in arithmetic series. The recursive formula is: [tex]a_{n} = a_{n-1} + d[/tex]

So, the recursive formula for given series is: [tex]a_{n} = a_{n-1} + 4[/tex]

Determining the Explicit Rule or Formula:

An explicit formula defines the nth term of the sequence, where n being the term's location. In other words, a sequence can be defined as a formula in terms of n. So,

First determine the sequence whether the sequence is in arithmetic. As we know that 2,6,10,14 is in arithmetic.
Then find the common difference. Here, d = 6 -2 = 10 - 6 = 14 - 10 = 4
Establishing an explicit formula by analyzing the pattern. i.e. adding first term to the product of d (common difference) and one less than the term number

Hence, the explicit formula is given by:

[tex]a_{n} = a_{1} + d (n-1)[/tex]

where,

aₙ is the nth term of the sequence

n is the term number

a₁ is the first term

d is the common difference

As the given sequence is: 2,6,10,14

a₁ = first term = 2

d = common difference = 6 - 2 = 4

Using explicit formula:

[tex]a_{n} = a_{1} + d (n-1)[/tex]

[tex]a_{n} = 10 + 5(n-1)[/tex]

[tex]a_{n} = 10 + 5n-5[/tex]

[tex]a_{n} = 5n+ 5[/tex]

Keywords: recursive formula, explicit formula, sequence

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