Question

Generating Patterns and illustrating Arithmetic Sequence


— Determine the corresponding nth term that generates the pattern of the following sequences. Provide complete solution in each item?

1, 7, 17, 33, ...​

Answer 1

Answer:

question 18, Laura found that the box is a little big to keep inside the locker. She realised the fact that a cubical box will exactly fit inside the locker. Find the smallest number by which 9000 must be divided so that the dimensions of the box form a perfect cube.

Answer 2

Given series: $1,7,17,31,------$

To find: the corresponding nth term that will generate the pattern of the following sequences.

Solution:

Observe the given series to find the pattern between the values and the place in the given sequence that the value has.

$\[\begin{array}{l}1 \to 1\\2 \to 7\\3 \to 17\\4 \to 31\end{array}\]$

Understand that, the value of the given sequence can be written as,

$\[\begin{array}{l}1 \to 1 = 2 \times \left( {{1^2}} \right) - 1\\2 \to 7 = 2 \times \left( {{2^2}} \right) - 1\\3 \to 17 = 2 \times \left( {{3^2}} \right) - 1\\4 \to 31 = 2 \times \left( {{4^2}} \right) - 1\end{array}\]$

Observe that in the above pattern the square of the positive integers (that increments with the next value) is multiplied by $2$ and then $1$ is subtracted from it to given the next consecutive value.

Therefore, it can be written in the general form as,

$\[n \to 2\left( {{n^2}} \right) - 1\]$, where $n=1,2,3,-------n$.