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AQA GCSE · Question 23.2 · Algebra
The nth term of a sequence is (2 + √3)ⁿ. Show that the third term is 26 + 15√3.
The nth term of a sequence is (2 + √3)ⁿ. Show that the third term is 26 + 15√3.
How to approach this question
1. To find the third term, substitute `n=3` into the expression for the nth term. This gives you (2 + √3)³.
2. To expand this, you can write it as (2 + √3)(2 + √3)(2 + √3).
3. First, expand the first two brackets. You can use the FOIL method (First, Outer, Inner, Last). Remember that √3 × √3 = 3.
4. Simplify the result of the first expansion.
5. Multiply this new expression by the final (2 + √3) bracket.
6. Collect like terms to get the final answer.
Full Answer
The third term is found by substituting n=3 into the formula.
Third term = (2 + √3)³
This can be written as (2 + √3)(2 + √3)(2 + √3).
Step 1: Expand the first two brackets.
(2 + √3)(2 + √3) = 2(2 + √3) + √3(2 + √3)
= 4 + 2√3 + 2√3 + (√3)²
= 4 + 4√3 + 3
= 7 + 4√3
Step 2: Multiply this result by the remaining (2 + √3).
(7 + 4√3)(2 + √3) = 7(2 + √3) + 4√3(2 + √3)
= 14 + 7√3 + 8√3 + 4(√3)²
= 14 + 15√3 + 4(3)
= 14 + 15√3 + 12
= 26 + 15√3
This is the required result.
We are given the formula for the nth term as (2 + √3)ⁿ. To find the third term, we need to set n = 3.
Third Term = (2 + √3)³
To evaluate this, we can expand the brackets.
(2 + √3)³ = (2 + √3)(2 + √3)(2 + √3)
First, let's expand (2 + √3)²:
(2 + √3)(2 + √3) = (2 × 2) + (2 × √3) + (√3 × 2) + (√3 × √3)
= 4 + 2√3 + 2√3 + 3
= 7 + 4√3
Now, we multiply this result by the remaining (2 + √3):
(7 + 4√3)(2 + √3) = (7 × 2) + (7 × √3) + (4√3 × 2) + (4√3 × √3)
= 14 + 7√3 + 8√3 + (4 × 3)
= 14 + 15√3 + 12
= 26 + 15√3
Thus, we have shown that the third term is 26 + 15√3.
Common mistakes
✗ Errors when expanding brackets, especially the surd terms. For example, forgetting the middle terms (2ab) when squaring a binomial, or calculating (√3)² as 9.
✗ Arithmetic errors when collecting like terms.
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