2(3) (11) 1. If P(2) 1.3 +2.4= = 6 2. Prove that P(n) is true for all positive integers n. and P(5) = 1.3+2 4+3.5+4.6+5.7= 5(6) (17) 6 what is P(n)?

For question number 22: Find the term iuicated in The (r + 1)st term of the expansion of (a + b)" is: (5x + 2) 5; Find the 5th term. O 160 O 1000x² O 200x O 400x he expansion (2) an-rbr. () an

(Зл — 1)! Simplify. 2!(3n + 1)!

Just need help with first two questions.

(п — 1)! (c)- (n + 2)! (n + 1)! (n – r + 1)! (d) (п —г — 1)! n! 6. Simplify: (a). n! (b)- (п — 3)!

Please explain thoroughly, I have observed every single explanation, but they lack information for a student to understand completely. I appreciate your effort.

Given that P(n) is the equation 1+3+5+7+...+(2n − 1) = n², where n is an integer such that n ≥ 1, we will prove that P(n) is true for all n ≥ 1 by induction. Base case: i. Write P(1). ii. Show that P(1) is true. In this case, this requires showing that a left-hand side is equal to a right-hand side. (b) Inductive hypothesis: Let k ≥ 1 be a natural number. Assume that P(k) is true. Write P(k). (c) Inductive step: i. Write P(k+1). ii. Use the assumption that P(k) is true to prove that P(k+1) is true. Justify all of your steps.

Give a combinatorial proof of the following identity: 2n (²2₂) = ²(2) + +n².

See attached image. Please provide solution with detailed explanation of steps.

Please help me solve this question