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Here is a sequence - OCR - GCSE Maths - Question 16 - 2018 - Paper 5
Question 16
Here is a sequence. 5 5 √3 15 15 √3 (a) Work out the next term. (b) Find the nᵗʰ term.
Worked Solution & Example Answer:Here is a sequence - OCR - GCSE Maths - Question 16 - 2018 - Paper 5
Step 1
(a) Work out the next term.
Answer
To find the next term in the sequence, let's first analyze the given terms:
- The first term is 5.
- The second term is 5√3.
- The third term is 15.
- The fourth term is 15√3.
We can observe a pattern based on alternating terms being multiplied:
- The odd-indexed terms (1st and 3rd) are: 5, 15. This follows a consistent pattern of multiplying by 3.
- The even-indexed terms (2nd and 4th) are: 5√3, 15√3. These terms also follow the same pattern of multiplying by 3.
Hence, the next term will be the 5th term, which is obtained by multiplying the previous odd-indexed term (15) by 3:
Next term = 15 × 3 = 45.
Step 2
(b) Find the nᵗʰ term.
Answer
In the sequence, we can denote the n-th term as follows:
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For odd n (1, 3, 5,...), the terms can be represented as: 5 imes 3^{ rac{n-1}{2}}
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For even n (2, 4, 6,...), the terms can be represented as: 5 imes 3^{ rac{n}{2}} imes ext{ }\sqrt{3}
Thus, we can combine these two patterns to express the n-th term in a concise manner, depending on whether n is odd or even:
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If n is odd: T_n = 5 imes 3^{ rac{n-1}{2}}
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If n is even: T_n = 5 imes 3^{ rac{n}{2}} imes ext{ }\sqrt{3}