 # Mathematics Questions

## Question

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## Transcribed Text

1. Prove the summation formula for all natural numbers n using induction: ∑ 1 𝑘(𝑘 + 1) 𝑛 𝑘=1 = 1 − 1 𝑛 + 1 2. Prove the inequality for all natural numbers n using induction: log2 𝑛 < n Hint: you may find the following identity helpful: 𝑛 + 1 = 𝑛(1 + 1 𝑛 ). You can use without proof that log2 𝑛 is a strictly increasing function. 3. Prove using structural induction that every member of the following recursively defined set 𝑆 has a remainder of 1 when divided by 5: 1 ∈ 𝑆 𝑛 ∈ 𝑆 → 5𝑛 + 1 ∈ 𝑆 𝑛 ∈ 𝑆 → 𝑛2 ∈ 𝑆 4. Find a closed-form representation of the following recursively defined sequence and show your work: 𝑎0 = 1, 𝑎1 = 2, 𝑎𝑛+2 = 𝑎𝑛+1 + 2𝑎𝑛 5. Find a closed-form representation of the following recursively defined sequence and show your work: 𝑎0 = 0, 𝑎1 = 0, 𝑎2 = −2, 𝑎3 = 0 𝑎𝑛+4 = −2𝑎𝑛+2 − 𝑎n

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