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Here is a sequence - OCR - GCSE Maths - Question 17 - 2020 - Paper 6
Question 17
Here is a sequence. 3 3√5 15 15√5 (a) Work out the next term. (b) Find the nth term.
Worked Solution & Example Answer:Here is a sequence - OCR - GCSE Maths - Question 17 - 2020 - Paper 6
Step 1
Work out the next term.
Answer
To find the next term in the sequence, we first need to observe the pattern:
- The first term is 3.
- The second term is 3√5.
- The third term is 15.
- The fourth term is 15√5.
We can see that the sequence alternates between multiplying the previous term by √5 and a constant factor of 5:
- From 3 to 3√5: multiply by √5.
- From 3√5 to 15: multiply by 5.
- From 15 to 15√5: multiply by √5.
Following this pattern, we multiply 15√5 by 5 to find the next term:
Next term = 15√5 × 5 = 75√5.
Step 2
Find the nth term.
Answer
To find the nth term, we recognize that:
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When n is odd, the term can be expressed as:
- For n = 1: 3 = 3 × 5^{(n-1)/2} = 3
- For n = 3: 15 = 3 × 5^{(n-1)/2} = 15
Therefore, for odd n:
nth term = 3 × 5^{(n-1)/2}
-
When n is even, the term is:
- For n = 2: 3√5 = 3 × 5^{n/2} = 3√5
- For n = 4: 15√5 = 3 × 5^{n/2} = 15√5
Thus, for even n:
nth term = 3 × 5^{n/2}.
Putting this together, we have:
- If n is odd: nth term = 3 × 5^{(n-1)/2}
- If n is even: nth term = 3 × 5^{n/2}.