R36 · Continue the sequence (doubling gaps)

Die Reihenfolge ist die der Primfaktoren von 1 = 1; 2 = 2; 4 = 2^2; 6 = 2 * 3; 8 = 2^3 und so weiter.

Die nächste Zahl ist also 2*5=10.

Die nächste Zahl in der Reihe ist **33**. Die Regel ist: Jede Zahl wird um das Doppelte des vorherigen Unterschieds erweitert.

**Schritt-by-Schritt Erklärung:**

1. Beginnen wir mit den gegebenen Zahlen: 2, 3, 5, 9, 17.
2. Berechnen die Differenzen zwischen aufeinanderfolgenden Termen:
   - 3 - 2 = 1
   - 5 - 3 = 2
   - 9 - 5 = 4
   - 17 - 9 = 8
3. Die Differenzen (1, 2, 4, 8) sind Potenzen von 2: 2⁰=1, 2¹=2, 2²=4, 2³=8.
4. Der nächste Unterschied wird also 2⁴=16 sein.
5. Addieren wir diesen Unterschied zur letzten Zahl hinzu:
   - 17 + 16 = 33
6. Daher ist die nächste Zahl **33**.

Die Regel: Jede Zahl ergibt sich aus der vorherigen Zahl plus dem Doppelten des letzten Differenzes, wobei der Unterschied selbst jeweils verdoppelt wird.

Die nächste Zahl ist **33**.

**Regel:** Die Differenz zwischen den aufeinanderfolgenden Zahlen verdoppelt sich bei jedem Schritt ($\text{+1, +2, +4, +8, und somit +16}$).

Die nächste Zahl ist **33**.

**Regel:** Die Differenz zwischen aufeinanderfolgenden Zahlen verdoppelt sich bei jedem Schritt ($+1, +2, +4, +8, \dots$).

Die nächste Zahl in der Reihe ist **33**.  
**Regel:** Jede Zahl entsteht durch Addition der doppelten Differenz des vorherigen Schritts.  
(2, 3, 5, 9, 17, **33**)  
Die Differenzen (1, 2, 4, 8, 16) verdoppeln sich jeweils, sodass die nächste Differenz 16 ist und 17 + 16 = 33 ergibt.

33.

**Regel:** Jede Zahl ist das Doppelte der vorherigen Zahl minus eins.

Die nächste Zahl ist **33**.

**Regel:** Die Differenz zwischen aufeinanderfolgenden Zahlen verdoppelt sich bei jedem Schritt ($+1$, dann $+2$, dann $+4$, dann $+8$, und somit wird als Nächstes $+16$ gerechnet: $17 + 16 = 33$).

Um die nächste Zahl zu finden, addiert man die beiden vorherigen Zahlen der Reihe und subtrahiert dann eins.

Die nächste Zahl ist also 28.

**Lösung:**

Die gegebene Zahlenreihe ist: **2, 3, 5, 9, 17, ?**

Um die Regel zu identifizieren, betrachten wir die Differenzen zwischen aufeinanderfolgenden Zahlen:

\[
\begin{align*}
3 - 2 &= 1 \\
5 - 3 &= 2 \\
9 - 5 &= 4 \\
17 - 9 &= 8 \\
\end{align*}
\]

Wir sehen, dass jede Differenz das **Doppelte** der vorherigen Differenz ist:

\[
\begin{align*}
1 \times 2 &= 2 \\
2 \times 2 &= 4 \\
4 \times 2 &= 8 \\
\end{align*}
\]

Folglich wird die nächste Differenz **\(8 \times 2 = 16\)** sein. Um die nächste Zahl in der Reihe zu erhalten, addieren wir diese Differenz zur letzten bekannten Zahl:

\[
17 + 16 = \boxed{33}
\]

**Regel:** Jede Zahl wird durch das Vorhergehende Plus das Doppelte des letzten Unterschieds berechnet.

Die nächste Zahl ist 33 - Die Reihe besteht aus den Hauptreihenfolgen der Fibonacci-Zahlen, wobei jede Zahl die Summe der beiden vorherigen ist.

<think>
Okay, let me see. The sequence is 2, 3, 5, 9, 17, and I need to find the next number. Hmm, so first I should look at the differences between the numbers. Let me subtract each number from the one that follows it.

Starting with 3 - 2 = 1. Then 5 - 3 = 2. Next, 9 - 5 = 4. After that, 17 - 9 = 8. Wait a second, those differences are 1, 2, 4, 8. That looks like they're doubling each time. So 1 times 2 is 2, then 2 times 2 is 4, 4 times 2 is 8. So if that pattern continues, the next difference should be 16, right? Because 8 times 2 is 16. 

So adding that to the last number in the sequence, which is 17, plus 16 would give 33. Let me check that again. Let's list the differences again to be sure: between 2 and 3 is +1, 3 to 5 is +2, 5 to 9 is +4, 9 to 17 is +8. Yep, each time the difference doubles. So the next difference is indeed 16, leading to 17 + 16 = 33. 

Wait, but just to make sure there isn't another pattern. Maybe it's something else? Let's think differently. Sometimes sequences can involve multiplying by a certain number and then adding or subtracting something. Let me try that approach. 

Starting with 2. If I multiply by 2, I get 4, but the next number is 3, which is 4 minus 1. Then 3 times 2 is 6, minus 1 would be 5, which matches. Then 5 times 2 is 10 minus 3 is 7? Wait no, the next number is 9. Hmm, maybe that's not the pattern. Alternatively, maybe each term is previous term times something plus something else.

Wait, let's see. From 2 to 3: 2 *1 +1=3. Then 3*1 +2=5, 5*1 +4=9, 9*1 +8=17. Ah, here the added numbers are again doubling each time: 1,2,4,8. So the multiplier is 1 each time, but adding the doubling numbers. So the next term would be 17 *1 +16=33. So same result as before. So that seems to confirm it.

Alternatively, another way to look at it: The pattern could be each term is previous term multiplied by 2 minus 1. Let's test that: 2*2-1=3, 3*2-1=5, 5*2-1=9, 9*2-1=17. Wait, that actually works too! Wait, hold on:

Wait, 2*2 is 4, minus 1 is 3. Then 3*2 is 6 minus 1 is 5, then 5*2 is 10 minus 1 is 9, then 9*2 is 18 minus 1 is 17. So that also gives the sequence correctly. So then following that rule, the next term would be 17*2 -1 = 34-1=33. So same answer again! So whether I see it as adding differences that double each time, or multiplying by 2 and subtracting 1 each time, both lead to 33. So that must be the right answer. 

So the rule is either "each term is the previous term multiplied by 2 minus 1" or "each term is the previous term plus 2 raised to (n-1) where n starts at 0". Since both explain the sequence and lead to the same next term, either explanation is valid, but the problem asks to explain it in one sentence. The simplest way to phrase it might be that each number is double the previous number minus 1. Let me check again with both methods:

Original sequence:

2

3 = 2*2 -1

5 = 3*2 -1

9 =5*2 -1

17=9*2 -1

So yes, that works perfectly. Then next is 17*2 -1=33. Alternatively, the difference approach: starting with 2, then each difference is 1,2,4,8,16,... which are 2^0, 2^1, 2^2, 2^3, 2^4 etc. So the nth term (starting at 0) is 2 + sum of 2^k from k=0 to n-1. Wait, let me see:

The first term is 2. Then the differences are 1 (2^0) added to get 3, then 2 (2^1) added to get 5, then 4 (2^2) to get 9, 8 (2^3) to get 17, so the next difference is 16 (2^4). So the nth term (starting at 1) is 2 + (2^1 -1). Wait, maybe another way. Alternatively, the formula for the nth term (if starting at term 0) would be 2^(n+1) -1. Let me check:

Term 0: 2^1 -1=1, no that's not matching. Wait, first term is 2. Hmm. Let me see:

Alternatively, the sequence is 2,3,5,9,17, which are each 1 more than a power of 2? Wait 3 is 2+1, 5 is 4+1, 9 is 8+1, 17 is 16+1. Wait a minute! 2 is 1+1, but 1 is 2^0. Then 3=2+1 (2^1+1?), but 2^1 is 2 plus 1 is 3. 5 is 4+1 (2^2+1), 9 is 8+1 (2^3+1), 17 is 16+1 (2^4+1). So following that pattern, the next term would be 2^5 +1 which is 32 +1=33. That's another way to see it. So each term is 2^(n) +1 where n starts at 1. Wait first term is 2, which would be 2^1 +0? Hmm, not exactly. Wait let's see the exponents:

Term 1: 2 = 2^1 + 0?

Term 2:3=2^1 +1

Term3:5=2^2 +1

Term4:9=2^3 +1

Term5:17=2^4 +1

So starting from the second term, each is 2^(n) +1 where n starts at 1. Wait the second term (3) is 2^1 +1, third term (5) is 2^2 +1, fourth (9) 2^3+1, fifth (17) 2^