19 (a) Here are the first four terms of a sequence - OCR - GCSE Maths - Question 19 - 2017 - Paper 1
Question 19
19 (a) Here are the first four terms of a sequence.
1 4 9 16
2 3 4 5
Find the $n$th term of this sequence.
(b) Here are the first four terms of a quadratic... show full transcript
Worked Solution & Example Answer:19 (a) Here are the first four terms of a sequence - OCR - GCSE Maths - Question 19 - 2017 - Paper 1
Step 1
Find the $n$th term of this sequence.
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Answer
The sequence given is 1, 4, 9, 16. Observing the values, they are perfect squares:
The first term is 12
The second term is 22
The third term is 32
The fourth term is 42
Thus, the general term for this sequence can be expressed as:
an=n2
Step 2
Find the values of $a$, $b$ and $c$.
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Answer
For the quadratic sequence given by the terms 2, 12, 28, 50, we observe the following:
First Differences:
12−2=10
28−12=16
50−28=22
Therefore, the first difference sequence is 10, 16, 22.
Second Differences:
16−10=6
22−16=6
The second differences are constant at 6, which suggests that the quadratic coefficient a = rac{6}{2} = 3.
To find b and c, we use the general form of the quadratic sequence an2+bn+c. Plugging in the first term:
For n=1:
3(12)+b(1)+c=2
This simplifies to:
ightarrow b + c = -1 ag{1}$$
For $n = 2:
3(22)+b(2)+c=12
Which simplifies to:
ightarrow 2b + c = 0 ag{2}$$
4. Now solving equations (1) and (2):
- From (1): $c = -1 - b$
- Substituting into (2):
$$2b - 1 - b = 0
ightarrow b - 1 = 0
ightarrow b = 1$$
- Now substituting $b$ back into (1):
$$1 + c = -1
ightarrow c = -2$$
Thus, the values are:
- $a = 3$
- $b = 1$
- $c = -2$
