Provide an appropriate response.(i) List the sequence of pentagonal numbers that corresponds to the sequence of dot patterns above and find two in addition to these. (ii) Complete the following list of equations suggested by the sequence of dot patterns and the pentagonal numbers you find in part (i). 1 = 1 1 + 4 = 5 1 + 4 + 7 = 12 1 + 4 + 7 + 10 = 22 _________________ = ___ _________________ = ___Observe that each pentagonal number is the sum of an arithmetic progression.(iii) Compute the eleventh term in the arithmetic progression 1, 4, 7, 10, ... .(iv) Compute the eleventh pentagonal number.(v) Determine a

Question

formula for nth term in the arithmetic progression 1, 4, 7, 10, ... .(vi) Determine a formula for the nth pentagonal number pn.

A. 1, 5, 7, 9, 11, 13, ...
(ii) 1 + 3 + 5 + 7 + 9 = 25
 1 + 3 + 5 + 7 + 9 + 11 = 36
(iii) Eleventh term: 1 + 3(11) = 34
(iv) Eleventh pentagonal number: 182
(v) nth term: 1 + 3(n) or 3n + 1
(vi) pn = 3n(n)/2
B. 1, 6, 18, 40, 76, 128, ...
(ii) 1 + 4 + 7 + 10 + 13 = 35
 1 + 4 + 7 + 10 + 13 + 16 = 51
(iii) Eleventh term: 3(10)- 1= 29
(iv) Eleventh pentagonal number: 220
(v) nth term: 1 + 3(n - 2) or 3n - 1
(vi) pn = 2n(n - 1)
C. (i) 1, 5, 12, 22, 35, 51, ...
(ii) 1 + 4 + 7 + 10 + 13 = 35
 1 + 4 + 7 + 10 + 13 + 16 = 51
(iii) Eleventh term: 1 + 3(10) = 31
(iv) Eleventh pentagonal number: 176
(v) nth term: 1 + 3(n - 1) or 3n - 2
(vi) pn = n(3n - 1)/2
D. 1, 5, 13, 23, 36, 52, ...
(ii) 1 + 4 + 7 + 10 + 13 = 35
 1 + 4 + 7 + 10 + 13 + 16 = 51
(iii) Eleventh term: 1 + 3(12) = 37
(iv) Eleventh pentagonal number: 187
(v) nth term: 1 + 3(n + 1) or 3n + 4
(vi) pn = n(3n + 1)/2

Answer 1

Answer: C