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This investigation looks at sequences of terms with square roots.
You can form a sequence by using square roots within square roots.
$$\sqrt{6}, \sqrt{6 + \sqrt{6}}, \sqrt{6 + \sqrt{6 + \sqrt{6}}}, \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6}}}}, \ldots$$
You can calculate the first four terms of this sequence as follows.
$$\sqrt{6} = \text{..................................} = 2.4494\ldots$$
$$\sqrt{6 + \sqrt{6}} = \sqrt{6 + 2.4494\ldots} = \sqrt{8.4494\ldots} = 2.9068\ldots$$
$$\sqrt{6 + \sqrt{6 + \sqrt{6}}} = \sqrt{6 + 2.9068\ldots} = \sqrt{8.9068\ldots} = 2.9844\ldots$$
$$\sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6}}}} = \sqrt{6 + 2.9844\ldots} = \sqrt{8.9844\ldots} = 2.9974\ldots$$
(a) Complete each part of the calculation below to work out the next term of the sequence, writing each decimal as far as the 4th decimal place.
$$\sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6 + \sqrt{6}}}}} = \sqrt{6 + \text{...................}} = \sqrt{\text{...............................}} = \text{............................}$$
(b) As the sequence continues, the terms get closer and closer to an integer. This integer is the \textit{integer limit} of the sequence.
Write down the integer limit of this sequence.
\text{.......................................................}
500
531
594
Here is a similar sequence of square roots. \( \sqrt{30}, \sqrt{30 + \sqrt{30}}, \sqrt{30 + \sqrt{30 + \sqrt{30}}}, \sqrt{30 + \sqrt{30 + \sqrt{30 + \sqrt{30}}}}, \ldots \)
(a) Calculate the first three terms, writing each decimal as far as the 4th decimal place.
\( .................................., \, .................................., \, .................................. \)
(b) Write down the integer limit of this sequence.
...............................................
539
516
505
(a) Complete this table for sequences similar to those in question 1(a) and question 2.
[Table_1:
\begin{array}{|c|c|}
\hline
\text{1st term} & \text{Integer limit} \\
\hline
\sqrt{2} & \\
\hline
\sqrt{6} & \\
\hline
\sqrt{12} & 4 \\
\hline
\sqrt{20} & \\
\hline
\sqrt{30} & \\
\hline
\sqrt{42} & 7 \\
\hline
\end{array}]
(b) (i) Use part (a) to find the first term of the sequence that has an integer limit of 8.
..............................................................
(ii) Calculate the 2nd term of the sequence in part (i), writing the decimal as far as the 4th decimal place.
..............................................................
552
547
512
The general sequence is $\sqrt{k}, \sqrt{k+\sqrt{k}}, \sqrt{k+\sqrt{k+\sqrt{k}}}, \sqrt{k+\sqrt{k+\sqrt{k+\sqrt{k}}}}, \ldots$
The integer limit of the sequence is the integer $N$.
For such sequences, $k = N(N-a)$, where $a$ is a constant.
(a) Use the last row of the table in question 3(a) to find the value of $a$.
..........................................
(b) Use $k = N(N-a)$ to show that $\sqrt{90}, \sqrt{90+\sqrt{90}}, \sqrt{90+\sqrt{90+\sqrt{90}}}, \sqrt{90+\sqrt{90+\sqrt{90+\sqrt{90}}}}, \ldots$
has an integer limit of $N = 10$.
(c) Find the first three terms, in square root form, of the sequence that has an integer limit of $N = 26$.
562
586
539
Here is the general form of another sequence of square roots with integer limit $N$.
$\sqrt{k}, \ \sqrt{k+2\sqrt{k}}, \ \sqrt{k+2\sqrt{k+2\sqrt{k}}}, \ \sqrt{k+2\sqrt{k+2\sqrt{k+2\sqrt{k}}}}, \ \ldots$
For such sequences, $k = N(N-a)$, where $a$ is a constant.
When $k = 24$ the integer limit of the sequence is $N = 6$.
(a) Find the value of the constant $a$.
..................................................
(b) (i) Find the value of $k$ when the integer limit is $N = 5$.
..................................................
(ii) Write down the 3rd term of the sequence in part (i) in square root form.
..................................................
(iii) Calculate the 3rd term, writing the decimal as far as the 4th decimal place.
..................................................
546
513
517
Here is the general form of another sequence of square roots with integer limit $N$.
$\sqrt{k}, \sqrt{k+5\sqrt{k}}, \sqrt{k+5\sqrt{k+5\sqrt{k}}}, \sqrt{k+5\sqrt{k+5\sqrt{k+5\sqrt{k}}}}, \ldots$
For this sequence, $k = N(N-a)$, where $a$ is a constant.
The sequence $\sqrt{14}, \sqrt{14+5\sqrt{14}}, \sqrt{14+5\sqrt{14+5\sqrt{14}}}, \sqrt{14+5\sqrt{14+5\sqrt{14}}}, \ldots$
has an integer limit of $N = 7$.
Find the value of the constant $a$.
537
530
548
Here is the general sequence with integer limit $N$.
$$\sqrt{k}, \; \sqrt{k + x\sqrt{k}}, \; \sqrt{k + x\sqrt{k + x\sqrt{k}}}, \; \sqrt{k + x\sqrt{k + x\sqrt{k + x\sqrt{k}}}}, \; \ldots$$
For all such sequences, $k = N(N-a)$, where $a$ is a constant that depends on the value of $x$.
(a) (i) Use question 4, question 5 and question 6 to complete the table.
$$\begin{array}{|c|c|} \hline x & a \\ \hline \text{Question 4} & 1 \\ \text{Question 5} & \\ \text{Question 6} & 5 \\ \hline \end{array}$$
(ii) Write down an expression for $k$ in terms of $N$ and $x$.
(b) Find the integer limit, $N$, of this sequence.
$$\sqrt{7}, \; \sqrt{7 + 6\sqrt{7}}, \; \sqrt{7 + 6\sqrt{7 + 6\sqrt{7}}}, \; \sqrt{7 + 6\sqrt{7 + 6\sqrt{7 + 6\sqrt{7}}}}, \; \ldots$$
582
508
546
