Derive a general formula for calculating the last term in a specific mathematical sequence | Step-by-Step Solution
Problem
Find a general formula for the sequence 1+3+5+7+9+13+17+21+25, exploring how to determine the last term in a sequence with 5 sides and developing an O(n) time solution
🎯 What You'll Learn
- Develop pattern recognition skills
- Learn systematic approach to sequence analysis
- Understand computational complexity optimization
Prerequisites: Algebraic manipulation, Sequence pattern recognition, Basic computational thinking
💡 Quick Summary
Hey there! This is a fantastic pattern recognition problem that combines sequence analysis with some geometric thinking. I can see you're working with a sequence that has some kind of grouping or "sides" structure, which suggests this might be related to a geometric pattern like a spiral or polygon. What do you notice when you look at the differences between consecutive terms in your sequence - are they constant throughout, or do you see some kind of grouping pattern? Also, think about what "5 sides" might mean in the context of this sequence - could it be telling you something about how the terms are organized or grouped together? I'd encourage you to write out the first several terms and calculate the differences between them, then see if you can spot how the sequence might be broken into distinct "sides" or groups. Once you identify that pattern, you can think about how each group relates to the next and build your general formula from there!
Step-by-Step Explanation
1. What We're Solving:
We need to find a general formula for the sequence 1+3+5+7+9+13+17+21+25 and figure out how to calculate the last term when we have a sequence with "5 sides" (which means 5 terms per side or group). We also want an efficient O(n) solution!2. The Approach:
This is a great detective problem. We need to look for patterns in the sequence. The key insight is that this isn't just a simple arithmetic sequence - there seems to be some grouping or geometric structure involved. Let's analyze the differences between terms and see if we can spot the underlying pattern.3. Step-by-Step Solution:
Step 1: Write out the sequence and find differences
- Sequence: 1, 3, 5, 7, 9, 13, 17, 21, 25
- First differences: 2, 2, 2, 2, 4, 4, 4, 4
- The pattern isn't uniform - we have groups!
- Four differences of 2: (1→3, 3→5, 5→7, 7→9)
- Four differences of 4: (9→13, 13→17, 17→21, 21→25)
- Side 1: 1, 3, 5, 7, 9 (5 consecutive odd numbers)
- Side 2: 13, 17, 21, 25 (4 terms with difference 4)
- Side 1 starts at: 1
- Side 2 starts at: 13
- The jump from end of side 1 (9) to start of side 2 (13) is 4
4. The Answer:
For a sequence with n "sides":- Side k starts at: 1 + 4k(k-1)
- Common difference in side k: 2k
- Last term of side k: [1 + 4k(k-1)] + (4)(2k) = 1 + 4k² + 4k = 4k² + 4k + 1
O(n) Solution: Simply plug k into the formula 4k² + 4k + 1 - this is O(1) per calculation!
5. Memory Tip:
Think of this as building a square spiral! Each "side" gets longer and the steps between points get bigger. The formula 4k² + 4k + 1 looks like it's related to square numbers - and that makes sense for a geometric pattern with "sides"!Great job tackling this complex pattern recognition problem! 🌟
⚠️ Common Mistakes to Avoid
- Focusing on individual terms instead of overall pattern
- Overlooking potential algorithmic solutions
- Failing to generalize the sequence formula
This explanation was generated by AI. While we work hard to be accurate, mistakes can happen! Always double-check important answers with your teacher or textbook.
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📷 Problem detected:
Solve: 2x + 5 = 13
Step 1:
Subtract 5 from both sides...
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