+1 vote
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Consider the function $h$ defined on $\{0,1,…….10\}$ with $h(0)=0, \: h(10)=10$ and

$$2[h(i)-h(i-1)] = h(i+1) – h(i) \: \text{ for } i = 1,2, \dots ,9.$$

Then the value of $h(1)$ is

1. $\frac{1}{2^9-1}\\$
2. $\frac{10}{2^9+1}\\$
3. $\frac{10}{2^{10}-1}\\$
4. $\frac{1}{2^{10}+1}$
in Calculus
edited | 357 views

If we see the sequence carefully , all it is saying is that common difference of the sequence is in Geometric progression with common ratio $2$ .

$h(0) = 0$ (given)

Let $h(1) = x$

$h(2) = x + 2x = 3x$

$h(3) = 3x + 4x = 7x$

and so on. The general term of the sequence is

$h(n) = (2^n -1) x$ where $x = h(1)$

so $h(n) = (2^n -1) h(1)$ and
$h(10) = 10$ . Substituting we get $C$ as the answer.
by Active (2k points)

+1 vote