Table of contents
- 0. Review of Algebra(0)
- 1. Equations & Inequalities(0)
- 2. Graphs of Equations(0)
- 3. Functions(0)
- 4. Polynomial Functions(0)
- 5. Rational Functions(0)
- 6. Exponential & Logarithmic Functions(0)
- 7. Systems of Equations & Matrices(0)
- 8. Conic Sections(0)
- 9. Sequences, Series, & Induction(0)
- 10. Combinatorics & Probability(0)
9. Sequences, Series, & Induction
Sequences
9. Sequences, Series, & Induction
Sequences: Study with Video Lessons, Practice Problems & Examples
61PRACTICE PROBLEM
Prove that the given statement is true for every positive integer n. Use mathematical induction.
![fraction numerator 1 space over denominator 1 times 3 end fraction plus fraction numerator space 1 space over denominator 2 times 4 end fraction plus fraction numerator space 1 space over denominator 3 times 5 end fraction plus space... space plus fraction numerator space 1 space over denominator n left parenthesis n plus 2 right parenthesis end fraction equals fraction numerator space n left parenthesis 3 n plus 5 right parenthesis over denominator 4 left parenthesis n plus 1 right parenthesis left parenthesis n plus 2 right parenthesis end fraction {"mathml":"<math style=\"font-family:stix;font-size:16px;\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><mstyle mathsize=\"16px\"><mfrac><mrow><mn>1</mn><mo> </mo></mrow><mrow><mn>1</mn><mo>·</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mo> </mo><mn>1</mn><mo> </mo></mrow><mrow><mn>2</mn><mo>·</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mo> </mo><mn>1</mn><mo> </mo></mrow><mrow><mn>3</mn><mo>·</mo><mn>5</mn></mrow></mfrac><mo>+</mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mo> </mo><mo>+</mo><mfrac><mrow><mo> </mo><mn>1</mn><mo> </mo></mrow><mrow><mi>n</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></mfrac><mo>=</mo><mfrac><mrow><mo> </mo><mi>n</mi><mo>(</mo><mn>3</mn><mi>n</mi><mo>+</mo><mn>5</mn><mo>)</mo></mrow><mrow><mn>4</mn><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></mfrac></mstyle></math>","truncated":false}](https://lightcat-files.s3.amazonaws.com/problem_images/fa6397dc5faace67-1671735422924.jpg)
Prove that the given statement is true for every positive integer n. Use mathematical induction.