Sigma Notation: Sum of Powers from \( r^1 \) to \( r^8 \)

Sigma notation is a concise way to represent the sum of a series. Below are the most important theorems and standard results involving powers of natural numbers, which frequently appear in exams like CAT, JEE, SSC, and CBSE board exams.

Basic Sigma Theorems

\( \sum_{r=1}^{n} (a_r \pm b_r) = \sum_{r=1}^{n} a_r \pm \sum_{r=1}^{n} b_r \)
\( \sum_{r=1}^{n} k \cdot a_r = k \sum_{r=1}^{n} a_r \)
\( \sum_{r=1}^{n} k = nk \)

where \(k\) is a constant

Sum of Powers of First \(n\) Natural Numbers

1. First Power: \( r^1 \)

\( \sum_{r=1}^{n} r = \frac{n(n+1)}{2} \)

2. Second Power: \( r^2 \)

\( \sum_{r=1}^{n} r^2 = \frac{n(n+1)(2n+1)}{6} \)

3. Third Power: \( r^3 \)

\( \sum_{r=1}^{n} r^3 = \left(\frac{n(n+1)}{2}\right)^2 \)

4. Fourth Power: \( r^4 \)

\( \sum_{r=1}^{n} r^4 = \frac{n(n+1)(2n+1)(3n^2 + 3n - 1)}{30} \)

5. Fifth Power: \( r^5 \)

\( \sum_{r=1}^{n} r^5 = \frac{n^2(n+1)^2(2n^2 + 2n - 1)}{12} \)

6. Sixth Power: \( r^6 \)

\( \sum_{r=1}^{n} r^6 = \frac{n(n+1)(2n+1)(3n^4 + 6n^3 - 3n + 1)}{42} \)

7. Seventh Power: \( r^7 \)

\( \sum_{r=1}^{n} r^7 = \frac{n^2(n+1)^2(3n^4 + 6n^3 - n^2 - 4n + 2)}{24} \)

8. Eighth Power: \( r^8 \)

\( \sum_{r=1}^{n} r^8 = \frac{n(n+1)(2n+1)(5n^6 + 15n^5 + 5n^4 - 15n^3 - n^2 + 3n - 1)}{90} \)

Why Learn These Formulas?

Knowing these formulas helps solve summation problems quickly during exams. They're particularly helpful for series simplification, definite integrals in calculus, and algebraic proofs.

Exam Relevance

CAT/CMAT/XAT: Arithmetic and algebra questions often test \( r^1 \) to \( r^3 \)
JEE: Up to \( r^6 \) may be required in coordinate geometry, sequences, and integration
CBSE/ISC Class 11-12: Part of NCERT syllabus under Mathematical Induction and Series

Final Thoughts

Whether you're preparing for a competitive exam or building a strong math foundation, mastering sigma notation and power sums will make you quicker, sharper, and more confident with series problems.

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