GCSE OCR Maths Sequences: Here are the first four terms of

Here are the first four terms of a sequence - OCR - GCSE Maths - Question 12 - 2019 - Paper 4

Question 12

Here are the first four terms of a sequence. -1 4 9 14 Write an expression for the $n$th term of this sequence. (b) The $n$th term of another sequence is give... show full transcript

Worked Solution & Example Answer:Here are the first four terms of a sequence - OCR - GCSE Maths - Question 12 - 2019 - Paper 4

Step 1

Write an expression for the $n$th term of this sequence.

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Answer

To find the expression for the $n$ th term of the sequence, we first observe the pattern in the given terms: -1, 4, 9, 14.

Calculate the differences between successive terms:

4 - (-1) = 5
9 - 4 = 5
14 - 9 = 5

The first differences are constant, indicating this is a linear sequence. We can express the $n$ th term in the form:

$t_n = an + b$

Using the known terms, we can set up the following equations:

For $n=1$ : $t_1 = a(1) + b = -1$ .
For $n=2$ : $t_2 = a(2) + b = 4$ .

Substituting from the first equation into the second, we find:

Step 2

Find the value of a and the value of b.

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Answer

We are given the expression for the $n$ th term as ( an^2 + bn ). To find the values of $a$ and $b$ , we can use the information provided:

The third term is equal to 9: $a(3^2) + b(3) = 9 \Rightarrow 9a + 3b = 9 \Rightarrow 3a + b = 3 \text{ (Equation 1)}$
The sixth term is equal to 126: $a(6^2) + b(6) = 126 \Rightarrow 36a + 6b = 126 \Rightarrow 6a + b = 21 \text{ (Equation 2)}$

Now, we can solve these equations simultaneously: Subtract Equation 1 from Equation 2: $(6a + b) - (3a + b) = 21 - 3 \Rightarrow 3a = 18 \Rightarrow a = 6.$ Now substituting $a = 6$ back into Equation 1: $3(6) + b = 3 \Rightarrow 18 + b = 3 \Rightarrow b = -15.$

Thus, the final values are:

$a = 6$
$b = -15$ .

Here are the first four terms of a sequence - OCR - GCSE Maths - Question 12 - 2019 - Paper 4

Worked Solution & Example Answer:Here are the first four terms of a sequence - OCR - GCSE Maths - Question 12 - 2019 - Paper 4

Write an expression for the $n$th term of this sequence.

Find the value of a and the value of b.

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