Solution Found!
Let P(n) be the statement that 13 + 23+ +n3(n + 1)/2)2 for
Chapter 5, Problem 4E(choose chapter or problem)
Let P(n) be the statement that 13 + 23+ ???+n3(n + 1)/2)2 for the positive integer n.a) What is the statement P(1)?________________b) Show that P(1) is true, completing the basis step of the proof.________________c) What is the inductive hypothesis?________________d) What do you need to prove in the inductive step?________________e) Complete the inductive step, identifying where you use the inductive hypothesis.
Questions & Answers
QUESTION:
Let P(n) be the statement that 13 + 23+ ???+n3(n + 1)/2)2 for the positive integer n.a) What is the statement P(1)?________________b) Show that P(1) is true, completing the basis step of the proof.________________c) What is the inductive hypothesis?________________d) What do you need to prove in the inductive step?________________e) Complete the inductive step, identifying where you use the inductive hypothesis.
ANSWER:Step 1 of 6
The statement P(1)
We have
![1 cubed plus 2 cubed plus..... plus n cubed equals open parentheses fraction numerator n open parentheses n plus 1 close parentheses over denominator 2 end fraction close parentheses squared](https://lh3.googleusercontent.com/PFt2QxhgX86T_DTMuv4bkb4NYxtV8dPi7Wsx5au29wcqwp3o59tVERgT4adkK_8kiPb8uvKSU8_jxeIdtguCaZyZh-Ga9j3TkhlFrj7DvjF_atxUR-CAwvpK9maqVMnHxsqm_ZpG2iPKGKp9Ao0 "{\"mathml\":\"<math style=\\"font-family:stix;font-size:16px;\\" xmlns=\\"http://www.w3.org/1998/Math/MathML\\"><mstyle mathsize=\\"16px\\"><msup><mn>1</mn><mn>3</mn></msup><mo>+</mo><msup><mn>2</mn><mn>3</mn></msup><mo>+</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>+</mo><msup><mi>n</mi><mn>3</mn></msup><mo>=</mo><msup><mfenced><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></mfenced><mn>2</mn></msup></mstyle></math>\"}")
Substitute n=1 in the above equation
To get
![1 cubed equals open parentheses fraction numerator 1 open parentheses 1 plus 1 close parentheses over denominator 2 end fraction close parentheses squared
P open parentheses 1 close parentheses colon 1 cubed equals open parentheses fraction numerator 1 open parentheses 1 plus 1 close parentheses over denominator 2 end fraction close parentheses squared](https://lh3.googleusercontent.com/5705sn838FRifzYiso5vG2qMP6nPcd2Ceeh44mcJBg6LWPWDXzty2NE0FdRx6LVw7adD7hA1KWBDbWgxQzz_iJEyqH8oCwMiLm27bFno5ZQBoCKBzKuoY6evEwoBwhEhaFWksETwmaryCy5mPDg "{\"mathml\":\"<math style=\\"font-family:stix;font-size:16px;\\" xmlns=\\"http://www.w3.org/1998/Math/MathML\\"><mstyle mathsize=\\"16px\\"><msup><mn>1</mn><mn>3</mn></msup><mo>=</mo><msup><mfenced><mfrac><mrow><mn>1</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mspace linebreak=\\"newline\\"/><mi>P</mi><mfenced><mn>1</mn></mfenced><mo>:</mo><msup><mn>1</mn><mn>3</mn></msup><mo>=</mo><msup><mfenced><mfrac><mrow><mn>1</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></mfenced><mn>2</mn></msup></mstyle></math>\"}")