Answer:
The given symbol for the nth triangular number = Tₙ
The given symbol for the nth square number = Qₙ
The given symbol for the nth pentagonal number = Pₙ
The mathematical formula for the nth triangular number, Tₙ = n·(n + 1)/2
The nth square number is given by Qₙ = n²
The nth pentagonal number is given by Pₙ = n·(3·n - 1)/2
(a) T₃.ₙ₋₁ = (3·n - 1)·(3·n - 1 + 1)/2 = (3·n - 1)·(3·n)/2 = 3·(3·n - 1)·(n)/2 = 3·Pₙ
Therefore;
T₍₃.ₙ ₋ ₁₎ = 3·Pₙ
(b) Pₙ - Qₙ = n·(3·n - 1)/2 - n²
Expanding the above equation gives;
n·(3·n - 1)/2 - n² = (3·n² - n - 2·n²)/2 = (n² - n)/2 = n·(n - 1)/2
T₍ₙ ₋ ₁₎ = ((n - 1)· ((n - 1) + 1))/2 = ((n - 1)· n)/2 = n·(n - 1)/2
∴ Pₙ - Qₙ = n·(3·n - 1)/2 - n² = n·(n - 1)/2 = T₍ₙ ₋ ₁₎
∴ Pₙ - Qₙ = T₍ₙ ₋ ₁₎
Therefore, we get;
P₃.ₙ - Q₃.ₙ = T₍₃.ₙ ₋ ₁₎
Where, T₍₃.ₙ ₋ ₁₎ = 3·Pₙ from (a) above, gives;
P₃.ₙ - Q₃.ₙ = 3·Pₙ
∴ P₃.ₙ - 3·Pₙ = Q₃.ₙ
Plugging in the values, gives;
P₃.ₙ = (3·n)·(3·(3·n) - 1)/2 = (3·n)·((9·n) - 1)/2
3·Pₙ = 3·n·(3·n - 1)/2
P₃.ₙ - 3·Pₙ = (3·n)·((9·n) - 1)/2 - 3·n·(3·n - 1)/2 = (3·n)·(((9·n) - 1) - (3·n - 1))/2
(3·n)·(((9·n) - 1) - (3·n - 1))/2 = (3·n)·(9·n - 1 - 3·n + 1)/2 = (3·n)·(6·n )/2 = 9·n²
Q₃.ₙ = (3·n)² = 9·n²
∴ P₃.ₙ - 3·Pₙ = P₃.ₙ - 3·Pₙ
Step-by-step explanation:
