Moderate -0.5 This is a straightforward proof requiring only the definition of a geometric sequence (u_n = ar^(n-1)) and the geometric mean formula. Students need to show that u_(n+1) = √(u_n × u_(n+2)), which follows directly by substituting ar^n, ar^(n+1), ar^(n+2) and simplifying. It's slightly easier than average because it's a direct application of definitions with minimal algebraic manipulation, though it does require understanding what 'geometric mean' means.
13 Consider a geometric sequence in which all the terms are positive real numbers. Show that, for any three consecutive terms of this sequence, the middle one is the geometric mean of the other two.
Expressions for three consecutive terms of a GP (any correct form)
The geometric mean of first and last is \(\sqrt{\frac{c}{r} \cdot cr}\)
M1
Expression for GM of first and last term (any correct form) FT their terms
\(\sqrt{\frac{c}{r} \cdot cr} = \sqrt{c^2} = c\); this is the middle term
E1
AG Correct completion
## Question 13:
| Answer | Mark | Guidance |
|--------|------|----------|
| Let the terms be $\frac{c}{r}$, $c$, $cr$ | B1 | Expressions for three consecutive terms of a GP (any correct form) |
| The geometric mean of first and last is $\sqrt{\frac{c}{r} \cdot cr}$ | M1 | Expression for GM of first and last term (any correct form) FT their terms |
| $\sqrt{\frac{c}{r} \cdot cr} = \sqrt{c^2} = c$; this is the middle term | E1 | AG Correct completion |
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13 Consider a geometric sequence in which all the terms are positive real numbers. Show that, for any three consecutive terms of this sequence, the middle one is the geometric mean of the other two.
\hfill \mbox{\textit{OCR MEI Paper 3 2018 Q13 [3]}}