Question

Write down the nth term of the following sequence

the following pattern are made using small squares

Answer 1

Explanation:

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Answer 2

a)
`n = 4 + 7\cdot i`,
`\forall\, i \in \mathbb{N}_O`, b)
`n = 2 + (i+1)^(2)`,
`\forall \,i\in \mathbb{N}_(O)`

Explanation:

a) The sequence is representative for an arithmetic sequence, whose key characteristic is that difference is between two consecutive elements is the same. In particular, the sequence has a difference of 7 between any two consecutive elements and the initial element is 4. Hence, we can derive the following formula:

`n = n_(o) + r\cdot i`,
`\forall\, i \in \mathbb{N}_O` (1)

Where:

`n_(o)` - Initial element.

`r` - Difference between two consecutive elements.

`i` - Index.

If we know that
`n_(o) = 4` and
`r = 7`, then the expression for the n-th term of the sequence is:

`n = 4 + 7\cdot i`,
`\forall \,i\in\mathbb{N}_(O)`

b) In this case, we have a geometric sequence described by the following equation:

`n = 2 + (i+1)^(2)`,
`\forall \,i\in \mathbb{N}_(O)` (2)

The constant element (
`2`) represents the two extreme squares, whereas the second order binomial represents the total of squares in the middle (
`(i+1)^(2)`) and emulates the area formula of the square.