Answer:
See below.
Step-by-step explanation:
The nth triangular number is Tn = n(n + 1)/2, the nth square number Qn = n^2 and the nth pentagonal number Pn = (3n^2 - n) / 2.
(a. Q6 + P5 = 6^2 + (3(5)^2 - 5)/2
= 36 + 35 = 71.
3T5 + Q5 + 1 = 3* 5(5 + 1) / 2 + 5^2 + 1
= 90 / 2 + 26
= 71.
So Q6 + P5 = 3T5 +Q5 +1.
(b) Qn+1 + Pn = (n + 1)^2 + (3n^2 - n) / 2
= n^2 + 2n + 1 + (3n^2 - n) / 2
= n^2 + 2n + 1 + 1.5n^2 - 0.5n
= 2.5n^2 + 1.5n + 1.
3Tn +Qn + 1 = 3n(n + 1) / 2 + n^2 + 1
= 3n^2 + 3n / 2 + n^2 + 1
= 1.5n^2 + 1.5n + n^2 + 1
= 2.5n^2 + 1.5n + 1.
Therefore they are equal for all positive n.
