07.04 Mathematical InductionAssignment Name: ___________________________________Print this page or copy and paste this assignment in to a word document.After filling in the requested information below, you may either scan your handwritten work or take a clear digital picture. You will need to save the scanned copy or digital picture of your work in order to submit as a file attachment.Note: When saving files, please remember that you should not use a period or a space in your file name. It is recommended that you use the underscore when saving your documents: 1_01, 1_02pg1, 1_02og2, etc. Contact your instructor if you have questions about saving files.Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false.(4 points each.)4 ⋅ 6 + 5 ⋅ 7 + 6 ⋅ 8 + ... + 4n( 4n + 2) = 12 + 42 + 72 + ... + (3n - 2)2 = For the given statement Pn, write the statements P1, Pk, and Pk+1.(2 points)2 + 4 + 6 + . . . + 2n = n(n+1)

Learn by DoingThis activity gives you practice explaining the different parts of the StatCrunch print-out for a hypothesis test for a population mean. You will review the meaning of standard error, T-scores and the P-value.Some features of this activity may not work well on a cell phone or tablet. We highly recommend that you complete this activity on a computer.Here are the directions, grading rubric, and definition of high-quality feedback for the discussion board exercises. A list of StatCrunch directions is provided at the bottom of this page.PromptIn a previous lab we tested the following hypotheses, H 0 : μ = 4.73 and H a : μ < 4.73 (where μ is the mean number of alcoholic drinks consumed by students at a liberal arts college).In a random sample of 75 students, the mean was 3.93 alcoholic drinks consumed in a week. Here is the StatCrunch output from the hypothesis test. μ : Mean of variable H 0 : μ = 4.73 H A : μ ≠ 4.73 Hypothesis test results Variable Sample Mean Std. Err. DF T-Stat P-value number of drinks per week 3.9333333 0.43592434 74 -1.8275343 0.0717 Using this context and the StatCrunch output, explain the meaning of each of the following. Std. Err (standard error)T-stat (T-score)P-value In a previous matched pairs lab we tested the following hypotheses, where μ is the mean of the differences in corn yield for a plot of land (regular seed minus kiln-dried): H0: µ = 0 and Ha: µ < 0. In a random sample of 11 seeds of each type, the mean of the differences in the sample was -33.7. Here is a StatCrunch print-out of the hypothesis test.Paired T hypothesis test: μ D = μ 1 − μ 2 : Mean of the difference between Regular seed and Kiln-dried seed H 0 : μ D = 0 H A : μ D < 0 Hypothesis test results Difference Mean Std. Err. DF T-Stat P-value Regular seed - Kiln-dried seed -33.727273 19.951346 10 -1.6904761 0.0609 Differences stored in column, Differences.Using this context and the StatCrunch output, explain the meaning of each of the following. Std. Err (standard error)T-stat (T-score)P-value.