The first three terms of a geometric progression are √5/2, 5/4, 5√5/8. Work out the next term.
How to approach this question
1. First, find the common ratio (r) of the geometric progression. You can do this by dividing the second term by the first term.
2. Remember that to divide by a fraction, you multiply by its reciprocal.
3. You may need to rationalise the denominator to simplify the common ratio.
4. Once you have the common ratio, multiply the third term by it to find the fourth term.
Full Answer
In a geometric progression, you multiply by a common ratio (r) to get from one term to the next.
To find r, divide any term by the previous term.
r = (5/4) / (√5/2)
r = (5/4) * (2/√5)
r = 10 / (4√5)
r = 5 / (2√5)
To rationalise the denominator: r = (5 * √5) / (2√5 * √5) = 5√5 / (2*5) = √5 / 2.
So the common ratio is √5 / 2.
To find the next term, multiply the third term by r:
Next term = (5√5 / 8) * (√5 / 2)
= (5 * √5 * √5) / (8 * 2)
= (5 * 5) / 16
= 25 / 16
A geometric progression is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
1. **Find the common ratio (r):**
r = (Term 2) / (Term 1)
r = (5/4) / (√5/2)
To divide by a fraction, we multiply by its reciprocal:
r = (5/4) × (2/√5)
r = 10 / (4√5) = 5 / (2√5)
To simplify this, we can write 5 as (√5)².
r = (√5)² / (2√5) = √5 / 2.
Let's check this with Term 3 / Term 2:
r = (5√5/8) / (5/4) = (5√5/8) × (4/5) = (20√5) / 40 = √5 / 2.
The common ratio is indeed r = √5 / 2.
2. **Find the next term (Term 4):**
Term 4 = Term 3 × r
Term 4 = (5√5 / 8) × (√5 / 2)
Term 4 = (5 × √5 × √5) / (8 × 2)
Since √5 × √5 = 5:
Term 4 = (5 × 5) / 16
Term 4 = 25 / 16.
Common mistakes
✗ Errors in dividing fractions.
✗ Errors when working with surds, especially rationalising the denominator or multiplying surds.
✗ Arithmetic mistakes.
Expert