Sequences and formulas

To work out the 5th term by repeating the rule, we begin with the given first term and apply the recurrence again and again until we reach $u_5$. This is part of Topic 8: Sequences, Formulae and Proportional Reasoning.

From the question, we know:

• The first term is $u_1=2$
• The rule for the next term is $u_{n+1}=2u_n-1$

Using the rule $u_{n+1}=2u_n-1$, plug in $u_1$ to get the next term:

$\begin{array}{rl}{u}_2& =2(u_1)-1\\ u_2& =2(2)-1\\ u_2& =3\end{array}$

Now substitute the value of $u_2$ into the same rule to calculate $u_3$:

$\begin{array}{rl}{u}_3& =2(u_2)-1\\ u_3& =2(3)-1\\ u_3& =5\end{array}$

Repeat the recurrence once more using $u_3$ to obtain $u_4$:

$\begin{array}{rl}{u}_4& =2(u_3)-1\\ u_4& =2(5)-1\\ u_4& =9\end{array}$

A calculator can help you apply the same operation several times quickly and reduces the chance of small number mistakes. In tests, it’s also useful for checking your repeated calculations.

Use $u_4$ in the recurrence to reach the required term $u_5$:

$\begin{array}{rl}{u}_5& =2(u_4)-1\\ u_5& =2(9)-1\\ u_5& =17\end{array}$

Compare what we found with the answer choices:

• Option C is 9, but that matches the 4th term ($u_4$), not the 5th.
• Option D is 33, which would come next as the 6th term ($2\times 17-1=33$).
• Option B is 15, and it does not match any term we calculated.

Following the recurrence step-by-step gives $u_5=17$.

This is a recursive sequence: each term depends on the previous one. The information given is:

• The initial term: $u_1=2$
• The recursive rule: $u_{n+1}=2u_n-1$
• The target: Find the 5th term, $u_5$.

On a Graphic Display Calculator (GDC), use the sequence (or recursion) feature. Input the recurrence and the initial value so the calculator can generate a table of terms automatically.

From the calculator’s table (or by letting it iterate), the terms are:

$\begin{array}{rl}{u}_1& =2\\ u_2& =2(2)-1=3\\ u_3& =2(3)-1=5\\ u_4& =2(5)-1=9\\ u_5& =2(9)-1=17\end{array}$

So when $n=5$, the table shows $17$.

Compare the calculator result with the options:

• Option C matches $u_4=9$, so it is not the 5th term.
• Options B and D are not values we get for the first five terms.

Because both the recursion table and the calculations show $u_5=17$, the correct choice is the option with 17.

We want $u_5$, so we start by using the given first term and the recurrence:

$\begin{array}{rl}{u}_1& =2\\ u_{n+1}& =2u_n-1\end{array}$

This tells us the process: double the current term, then subtract 1 to get the next term.

Compute a few terms in order to see what the sequence looks like:

$\begin{array}{rl}{u}_1& =2\\ u_2& =2(2)-1=3\\ u_3& =2(3)-1=5\\ u_4& =2(5)-1=9\\ u_5& =2(9)-1=17\end{array}$

Looking at $2,3,5,9,17,\dots$, each term can be written as “a power of 2 plus 1”:

$\begin{array}{rl}{u}_1& =2^0+1=1+1=2\\ u_2& =2^1+1=2+1=3\\ u_3& =2^2+1=4+1=5\\ u_4& =2^3+1=8+1=9\\ u_5& =2^4+1=16+1=17\end{array}$

So a general rule that fits is $u_n=2^{n-1}+1$.

Using the values we listed:

• Option B is $15$, which is not in the sequence.
• Option C is $9$, but that is the $4$th term ($u_4$), not the $5$th.
• Option D is $33$, which would be the next term after 17 since $u_6=2(17)-1=33$.

Since the 5th term we found is $17$, the correct option is A.

In mathematics, specifically within the study of Sequences, Formulae and Proportional Reasoning, a sequence is defined like a function but with a restricted input set: its domain is the positive integers $\{1,2,3,\dots \}$.

A linear function $f(x)=ax+b$ can take any real input $x$ (such as $1.5$, $\pi$, or $-2$). But in a linear sequence $u_n=an+b$, the value of $n$ is an index (a term number in an ordered list), so it is only allowed to be a positive integer.

A quick way to compare is to graph both ideas. If you graph $f(x)=2x+1$, you see a continuous straight line. But if you graph the sequence $u_n=2n+1$, you only see separated dots at $n=1,n=2,n=3,\dots$.

Because a sequence only exists at these separate integer inputs, there is no defined term sitting “between” $n=1$ and $n=2$, so using $n=1.5$ is not valid for a sequence term.

Option D is based on the claim that $u_{1.5}$ should exist and be the average of $u_1$ and $u_2$. It’s true you can compute an average of two numbers, but that computed value does not become an official sequence term.

The index $n$ describes an actual position (1st, 2nd, 3rd, ...), and there is no “1.5th” position in a list.

Option C suggests the statement is only true when $a$ and $b$ are integers. But the key restriction for sequences is about the input value $n$, not about whether the constants or the outputs are whole numbers.

Even if $a$ and $b$ are not integers, the index still must be $1,2,3,\dots$, so this option adds an unnecessary condition.

Since a sequence is defined only for positive integer indices, the statement in the question matches the definition and is correct. The option claiming it is false must be rejected because it conflicts with how sequences are defined.

In mathematics, specifically within the topic of Sequences, Formulae and Proportional Reasoning, a sequence is treated as a function whose allowed inputs are only positive integers ($n=1,2,3,\dots$).

Even though $u_n=an+b$ looks the same as $f(x)=ax+b$, the variable $n$ is a term number. Just like you cannot stand in the “1.5th” place in a queue, you cannot label a sequence term with $n=1.5$.

Option D says $u_{1.5}$ is defined as the arithmetic mean of the first two terms.

You can certainly find the average of $u_1$ and $u_2$, but that result is not automatically part of the sequence. Sequence notation $u_n$ is only used when $n$ is an integer, so this option is not valid.

Option C claims the statement only works if $a$ and $b$ are integers.

But $a$ and $b$ can be any real numbers (for example, $u_n=0.5n+\pi$). The “must be a positive integer” rule applies to the index $n$ only, not to the constants or the sequence values. So Option C is incorrect.

Because the definition of a sequence requires discrete integer inputs ($n=1,2,3,\dots$), the statement given in the question is accurate. Choosing “False” would go against the basic definition taught for sequences.

A function can be drawn as an unbroken line because it accepts all real inputs. A sequence, however, appears as separated points because it is only defined at specific integer inputs.

This picture helps explain why there is no legitimate term at a non-integer index like $1.5$.

After eliminating the incorrect statements and using the domain rule for sequences, the correct choice is confirmed.

The sequence begins at index 0, so the term we start from is $u_0$. This value is provided directly.

$u_0=5$

The recursive rule says each new term is found by adding 3 to the previous term:

$u_{n+1}=u_n+3$

To get $u_1$, set $n=0$:

$u_1=u_0+3$ $u_1=5+3=8$

Now use the same rule starting from $u_1=8$. With $n=1$, the third term is:

$u_2=u_1+3$ $u_2=8+3=11$

Compare what each option represents:

• Option A is $5$, which corresponds to $u_0$ (the first listed term).
• Option D is $8$, which corresponds to $u_1$ (the second listed term).
• Option B is $14$, which would come next as $u_3$ because $11+3=14$.

So the third term in the sequence, $u_2$, is $11$, which matches Option C.

We are told $u_0=5$ and the recursion is $u_{n+1}=u_n+3$. Because the same number (3) is added each time, the sequence is arithmetic.

So we identify:

• The term at index 0: $u_0=5$
• The common difference: $d=3$

A common arithmetic-sequence formula is:

$\begin{array}{rl}{u}_n& =u_1+(n-1)d\end{array}$

But since this sequence starts at index 0, we rewrite it as:

$\begin{array}{rl}{u}_n& =u_0+n\cdot d\end{array}$

The question asks for the third term listed, which is $u_2$.

We can discard some answers immediately:

• Option A is 5, which is $u_0$, not $u_2$.
• Option D is 8, and checking $u_1=5+3=8$ shows it is the second term, not the third.

Apply $u_n=u_0+n\cdot d$ with $n=2$:

$\begin{array}{rl}{u}_2& =5+2(3)\\ u_2& =5+6\\ u_2& =11\end{array}$

With $u_2=11$, we can also rule out Option B (14), since that would be $u_3$ ($11+3=14$).

The calculation gives $u_2=11$, so the correct choice is the option showing 11.

The description is about beginning with an initial term and then using a recursive rule (where each term is built from the previous one) again and again to produce the next terms of a sequence.

This idea appears frequently in Sequences, Formulae and Proportional Reasoning.

Differentiation belongs to calculus and is used to calculate a function’s rate of change (its gradient) at a point. It is not about repeatedly using a recursive relationship to list terms.

Interpolation is used to estimate missing values between known data points. It is an estimation method, not a repeated rule for generating a sequence from a starting value.

Extrapolation means predicting values beyond the known data range by extending a trend. Even though it looks like “going forward,” it is a prediction technique, not a recursive term-by-term construction.

The correct word is iteration. Iteration means repeating the same procedure, where each output becomes the next input. In sequences/algorithms, applying a rule again and again (such as $u_{n+1}=f(u_n)$) is exactly iteration.

We are being asked to name a particular mathematical procedure. The important parts are:

1. "Repeatedly applying a recursive rule": the same rule is used over and over.
2. "Generate successive terms": we are building a sequence one term at a time.
3. "From a starting value": there is an initial term that begins the process.

This is central to Sequences, Formulae and Proportional Reasoning in the course.

Consider Option B.

Differentiation is used to find the rate of change (slope/gradient) of a function, often written using derivatives like $f^{\prime }(x)$. It does not describe repeatedly generating new terms from a previous term, so it cannot be correct.

Now look at Options C and D.

• Interpolation estimates a value that lies between known data points.
• Extrapolation predicts a value outside the known data range by extending a pattern or trend.

Both are about estimating from data, not about repeatedly feeding one term into a recursive rule to create the next term.

After removing the other choices, Option A remains.

Iteration is the correct term: it means repeating a rule where each result becomes the next input.

For instance, starting from $x_0$ and using $x_{n+1}=2x_n+1$, finding $x_1,x_2,x_3...$ is an example of iteration.