Question

101*10001*100000001*.................*1000..........1 =? 2n-1 Как решить???

Answer 1

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Answer 2

To solve the expression $101 \times 10001 \times 100000001 \times \ldots \times 1000 \ldots 1 = ?$ and express the result as $2^n-1$, we need to understand the pattern of the numbers being multiplied.

Let's observe the pattern:

$101 = 102 - 1$ $10001 = 10002 - 1$ $100000001 = 100000002 - 1$ $\ldots$ In general, for any number $n$, we can express $n$ as $n+1 - 1$.

So, the expression becomes:

$101 \times 10001 \times 100000001 \times \ldots \times 1000 \ldots 1$

$= (102 - 1) \times (10002 - 1) \times (100000002 - 1) \times \ldots \times (1000\ldots 0 - 1) \times (1 - 1)$

$= (102 \times 10002 \times 100000002 \times \ldots \times 1000\ldots 0) - (1 + 1 + 1 + \ldots + 1 + 1)$

The second part of the expression is the subtraction of a bunch of ones from the multiplication. We can simplify it further:

$1 + 1 + 1 + \ldots + 1 + 1 = k \times 1 = k$

where $k$ is the number of terms in the multiplication. As we can see, there are $k$ terms being multiplied together.

Now, we just need to find the value of $102 \times 10002 \times 100000002 \times \ldots \times 1000\ldots 0$.

Notice that each term in the multiplication is $2$ more than the previous term:

$102 = 100 + 2$ $10002 = 10000 + 2$ $100000002 = 100000000 + 2$

And so on.

So, the multiplication becomes:

$(100 + 2) \times (10000 + 2) \times (100000000 + 2) \times \ldots \times (1000\ldots 0 + 2)$

$= (100 \times 10000 \times 100000000 \times \ldots \times 1000\ldots 0) + (2 + 2 + 2 + \ldots + 2 + 2)$

Again, the second part of the expression is just $k \times 2 = 2k$, where $k$ is the number of terms in the multiplication.

Now, we have:

$102 \times 10002 \times 100000002 \times \ldots \times 1000\ldots 0 = (100 \times 10000 \times 100000000 \times \ldots \times 1000\ldots 0) + 2k$

But wait! The first part of this expression is simply:

$100 \times 10000 \times 100000000 \times \ldots \times 1000\ldots 0 = 100^{k}$

So, we have:

$102 \times 10002 \times 100000002 \times \ldots \times 1000\ldots 0 = 100^{k} + 2k$

Now, the final expression is:

$101 \times 10001 \times 100000001 \times \ldots \times 1000 \ldots 1 = (100^{k} + 2k) - k$

$= 100^{k} + k$

Now, we need to express this result in the form $2^n - 1$. Notice that $100 = 2^2$. So, we can rewrite the expression as:

$100^{k} + k = (2^2)^k + k$

Using the property $a^{mn} = (a^m)^n$, we get:

$(2^2)^k + k = 2^{2k} + k$

Now, we have the expression in the form $2^n + k$.

Since we want to express this in the form $2^n - 1$