l. [ Maximum Marks: 24] This question asks you to investigate pentagonal numhers to derive and prove its general term. Pentagonal timbers can he represented hjr dots as shown below where Pa denotes the nth pentagonal number. the T a! PI P3 5 dots are needed to create a regular pentagon with side of length I as shown in the second term. 12 dots are needed to create a regular pentagon with side of length 2 as shown in the third term. The length of the sides of the pentagon is increased to obtain the nest terms successively. {a} Write down the value ofps. [l] [b] By Ioolring for the patterns. show that p\" = p114 + [3n - ELI! E 3".\" E 2. [3] {c} Byespressingp.I asascriea. showtliatlrll,I = 33:1.\" E2\". [3] [d] Hence. determine whether 2'3le is a pentagonal number. [3] [c] Find the least pentagonal number which is greater than lt}. [5] {1] Consider the statement: 22 is the only:r pentagonal number which is divisible by l I. Show that this statement is false. [2] Sharon claims that given p1 = 1 and p11 = nu.1 + (3n - 2).n E 2'21: 2 2. [g] Show. by principle of mathematical induction. that Sharon's claim is true for all n E Z". [T]