Question

the first four terms of a sequence are 0, 3, 12 and 27
a) write down the 5th term 1
b) what is the expression for the nth term? 3
c) calculate
i. the value of the 20th term 2
ii. the value of n for which the nth term is 575 3

Explanation:

Analyze the given sequence terms

The first four terms of the sequence are:

$$ a_1 = 0,\quad a_2 = 3,\quad a_3 = 12,\quad a_4 = 27 $$

Let's look at the differences between consecutive terms:

• First differences:

$$ a_2 - a_1 = 3 - 0 = 3 $$

$$ a_3 - a_2 = 12 - 3 = 9 $$

$$ a_4 - a_3 = 27 - 12 = 15 $$

• Second differences:

$$ 9 - 3 = 6 $$

$$ 15 - 9 = 6 $$

Since the second differences are constant, this is a quadratic sequence.

Find the expression for the nth term

A quadratic sequence has the general form:

$$ a_n = an^2 + bn + c $$

The constant second difference is equal to \(2a\):

$$ 2a = 6 \implies a = 3 $$

Now we subtract \(3n^2\) from each term to find the remaining linear part:

• For \(n = 1\): \(0 - 3(1)^2 = -3\)
• For \(n = 2\): \(3 - 3(2)^2 = 3 - 12 = -9\)
• For \(n = 3\): \(12 - 3(3)^2 = 12 - 27 = -15\)
• For \(n = 4\): \(27 - 3(4)^2 = 27 - 48 = -21\)

The remaining sequence is \(-3, -9, -15, -21, \dots\), which is an arithmetic sequence with a first term of \(-3\) and a common difference of \(-6\).
Thus, the linear part \(bn + c\) is:

$$ bn + c = -6n + 3 $$

Let's combine these parts to get the general formula:

$$ a_n = 3n^2 - 6n + 3 $$

We can factor this expression:

$$ a_n = 3(n^2 - 2n + 1) = 3(n - 1)^2 $$

Let's verify this formula with the given terms:

• For \(n = 1\): \(3(1 - 1)^2 = 0\) (Correct)
• For \(n = 2\): \(3(2 - 1)^2 = 3\) (Correct)
• For \(n = 3\): \(3(3 - 1)^2 = 12\) (Correct)
• For \(n = 4\): \(3(4 - 1)^2 = 27\) (Correct)

Determine the 5th term

Using our verified formula \(a_n = 3(n - 1)^2\) for \(n = 5\):

$$ a_5 = 3(5 - 1)^2 = 3(4)^2 = 3 \times 16 = 48 $$

Calculate the value of the 20th term

Using the formula for \(n = 20\):

$$ a_{20} = 3(20 - 1)^2 = 3(19)^2 $$

First, calculate \(19^2\):

$$ 19^2 = 361 $$

Now, multiply by 3:

$$ a_{20} = 3 \times 361 = 1083 $$

Find the value of n when the term is 575

We set the general term formula equal to 575 and solve for \(n\):

$$ 3(n - 1)^2 = 575 $$

Divide both sides by 3:

$$ (n - 1)^2 = \frac{575}{3} \approx 191.67 $$

Since \(n\) must be an integer, let's re-examine the question text. The handwritten text says "575". Let's check if there is a typo in the question or if we should solve it algebraically:

$$ n - 1 = \sqrt{\frac{575}{3}} \approx 13.84 \implies n \approx 14.84 $$

If the value was intended to be \(507\) instead of \(575\):

$$ 3(n - 1)^2 = 507 \implies (n - 1)^2 = 169 \implies n - 1 = 13 \implies n = 14 $$

If the value is strictly \(575\) as written:

$$ n = 1 + \sqrt{\frac{575}{3}} \approx 14.84 $$

Let's provide the exact algebraic solution for the written value \(575\).

Answer:

Question a

The 5th term is <blank>48</blank>.

Question b

The expression for the \(n\)th term is <blank>\(3(n - 1)^2\)</blank>.

Question c

Part i

The value of the 20th term is <blank>1083</blank>.

Part ii

The value of \(n\) for which the \(n\)th term is 575 is <blank>\(1 + \sqrt{\frac{575}{3}}\)</blank> (approximately \(14.84\)).