Second sum ($k = 4$ to $8$):

["# Understanding the Second Sum $S_k = \sum_{i=4}^{k} (i + k)$: A Detailed Breakdown", "When tackling problems in discrete mathematics, summation notation plays a critical role in simplifying complex expressions. One such sum frequently encountered in algorithm analysis and sequence-based problems is the Second Sum defined as:", "$$\nS_k = \sum_{i=4}^{k} (i + k)\n$$", "This article explores the second sum $S_k$ from $i = 4$ to $i = k$, particularly for $k$ ranging from 4 to 8, providing a clear mathematical derivation, real-world relevance, and practical applications.", "---", "## What Is the Second Sum $\sum_{i=4}^{k} (i + k)$?", "The notation $\sum_{i=4}^{k} (i + k)$ means summing the expression $(i + k)$ for all integer values of $i$ starting at 4 and ending at $k$. This is a double-layer summation combining both $i$ and the fixed term $k$.", "Expanding the summation:", "$$\nS_k = \sum_{i=4}^{k} i + \sum_{i=4}^{k} k\n$$", "Since $k$ is constant with respect to $i$, the second term simplifies:", "$$\n\sum_{i=4}^{k} k = k \cdot (k - 4 + 1) = k(k - 3)\n$$", "The first term is the sum of integers from 4 to $k$:", "$$\n\sum_{i=4}^{k} i = \sum_{i=1}^{k} i - \sum_{i=1}^{3} i = \frac{k(k+1)}{2} - \frac{3 \cdot 4}{2} = \frac{k(k+1)}{2} - 6\n$$", "Putting it all together:", "$$\nS_k = \left( \frac{k(k+1)}{2} - 6 \right) + k(k - 3) = \frac{k(k+1)}{2} - 6 + k^2 - 3k\n$$", "Combine terms:", "$$\nS_k = \frac{k^2 + k - 12 + 2k^2 - 6k}{2} = \frac{3k^2 - 5k - 12}{2}\n$$", "So, the closed-form formula is:", "$$\n\boxed{ S_k = \frac{3k^2 - 5k - 12}{2} }\n$$", "---", "## Calculating $S_k$ for $k = 4$ to $k = 8$", "Let’s compute $S_k$ step by step using the formula and verify with direct summation:", "| $k$ | $S_k = \sum_{i=4}^{k} (i + k)$ | Closed-form Result $ \frac{3k^2 - 5k - 12}{2} $ |\n|-----|-------------------------------|---------------------------------------------------|\n| 4 | $ (4+4) + (4+4) = 8 + 8 = 16 $ | $\frac{3(16) - 20 - 12}{2} = \frac{48 - 32}{2} = 8 $ ❌ (inconsistency noted) |\n| | | Wait: direct sum: $ \sum_{i=4}^4 (i + 4) = 4 + 4 = 8 $ ✅ |", "Correction: There’s a small error — summing $i + k$ from $i=4$ to $k$ includes $k$ as a constant per term, but the formula derivation is correct. Let’s verify with actual computation:", "Direct Computation:", "- $k = 4$:\n $$\n S_4 = (4+4) + (5+4) + (6+4) + (7+4) + (8+4) \ ext{? No! Wait: } i \ ext{ runs from } 4 \ ext{ to } k = 4 \Rightarrow \ ext{only } i=4\n $$\n So $S_4 = (4 + 4) = 8$\n (Note: Typical range $i=4$ to $k$, so only one term when $k=4$)", "- $k = 5$: $i = 4,5$ → $ (4+5)+(5+5) = 9 + 10 = 19 $", "- $k = 6$: $i = 4,5,6$ → $8+6 + 9+6 + 10+6 = 14 + 15 + 16 = 45$", "- $k = 7$: $i = 4$ to $7$ → $8+7 + 9+7 + 10+7 + 11+7 = 15 + 16 + 17 + 18 = 66$", "- $k = 8$: $i = 4$ to $8$ → sum of $i + 8$:\n $12 + 13 + 14 + 15 + 16 + 17 + 18 = 105$", "Now compute closed-form result:", "- $S_4 = \frac{3(16) - 20 - 12}{2} = \frac{48 - 32}{2} = 8$ ✅\n- $S_5 = \frac{3(25) - 25 - 12}{2} = \frac{75 - 37}{2} = \frac{38}{2} = 19$ ✅\n- $S_6 = \frac{3(36) - 30 - 12}{2} = \frac{108 - 42}{2} = 33$ ❌ vs 45 — discrepancy!", "Wait — error in manual sum? Let’s recalculate $S_6$:", "$i=4$: $4 + 6 = 10$\n$i=5$: $5 + 6 = 11$\n$i=6$: $6 + 6 = 12$\nSum: $10 + 11 + 12 = 33$ ✅ matches formula.", "Earlier miscalculation was wrong. Correct manual sum: only three terms.", "Similarly:", "- $k=7$: $i=4$ to 7:\n $4+7=11$, $5+7=12$, $6+7=13$, $7+7=14$ → sum = $11+12+13+14 = 50$ ❌\nWait — only $i=4$ to $7$: 4 terms → but 7 - 4 + 1 = 4 terms: yes\nBut formula gives 66 — inconsistency.", "Wait: formula must be rechecked.", "---", "## Re-evaluating the Closed-Form Expression", "Go back:", "$$\nS_k = \sum_{i=4}^{k} (i + k) = \sum_{i=4}^{k} i + \sum_{i=4}^{k} k = \left( \sum_{i=1}^{k} i - \sum_{i=1}^{3} i \right) + (k - 4 + 1)k\n$$", "$$\n= \left( \frac{k(k+1)}{2} - 6 \right) + k(k - 3)\n$$", "$$\n= \frac{k(k+1)}{2} - 6 + k^2 - 3k\n$$", "$$\n= \frac{k^2 + k}{2} + k^2 - 3k - 6 = \frac{3k^2 + k - 6k - 12}{2} = \frac{3k^2 - 5k - 12}{2}\n$$", "Formula is correct.", "But manual for $k=7$:", "$$\n\sum_{i=4}^{7} (i + 7) = (4+7)+(5+7)+(6+7)+(7+7) = 11+12+13+14 = 50\n$$", "Formula: $\frac{3(49) - 35 - 12}{2} = \frac{147 - 47}{2} = 100/2 = 50$ ✅", "Earlier mistake: manual sum incorrectly included $i+k$ as three terms but miscalculated: 4 terms, not 4 — $i=4,5,6,7$ → 4 terms. Sum = 50, matches.", "Similarly:", "- $k=8$: $i=4$ to $8$: 5 terms\nFormula: $\frac{3(64) - 40 - 12}{2} = \frac{192 - 52}{2} = 140/2 = 70$\nManual: $8+8=16$, $9+8=17$, $10+8=18$, $11+8=19$, $12+8=20$ → sum $16+17+18+19+20 = 90$ — inconsistency!", "Wait: $i=4$: $4 + 8 = 12$, not 16!", "Previous manual sum flawed.", "Correct:", "$i+8$, $i=4$: $12$, $i=5$: $13$, $i=6$: $14$, $i=7$: $15$, $i=8$: $16$ → sum: $12+13+14+15+16 = 70$ ✅ matches formula.", "So corrected values:", "| $k$ | Direct Sum $| Formula $S_k = \frac{3k^2 - 5k - 12}{2}$ |\n|-----|-------------|---------------------------------------------|\n| 4 | 8 | 8 |\n| 5 | 19 | 19 |\n| 6 | 33 | 33 |\n| 7 | 50 | 50 |\n| 8 | 70 | 70 |", "---", "## Why Is This Sum Important?", "### 1. Algorithm Time Complexity", "This sum arises naturally when analyzing the total runtime or operations in algorithms involving nested loops. For instance, consider a pseudo-code block:", "python\nfor i from 4 to k: \n for j from 4 to k: \n  process(i, k)", "Even if inner loop runs similarly, sums involving $i + k$ per iteration decompose neatly using this formula.", "### 2. Arithmetic Series Exploration", "Breaking $S_k = \sum(i + k) = \sum i + \sum k$ illustrates the distributive property of summation — key for transforming recursive relations into closed-form expressions.", "### 3. Educational Tool", "Teaching this sum helps reinforce:\n- Summation notation\n- Linear summation identities\n- Closed-form derivation via decomposition", "---", "## Final Thoughts", "The second sum $S_k = \sum_{i=4}^{k} (i + k)$ is a foundational example of structured summation with practical applications in algorithm analysis and discrete math. Understanding its derivation and verifying via both direct and closed-form methods ensures fluency in mathematical reasoning—essential for coding challenges, algorithm design, and academic success.", "---", "## Summary Table", "| $k$ | $S_k = \sum_{i=4}^{k} (i + k)$ | Computation Method |\n|-----|-------------------------------|-------------------------------|\n| 4 | 8 | Direct sum or formula |\n| 5 | 19 | Formula or step-by-step |\n| 6 | 33 | Formula or direct sum |\n| 7 | 50 | Formula or direct sum |\n| 8 | 70 | Formula or direct sum |", "---", "## References & Further Reading", "- Summation formulas in discrete mathematics\n- Algorithm analysis and big-O notation\n- Derivation of arithmetic series formulas\n- Applications of summation in computer science", "Start mastering sums like $S_k$ today — they unlock deeper insights into computation and logic!"]