## Thinking Mathematically (6th Edition)

a. $28, 36, 45, 55, 66$ b.$36, 49, 64, 81, 100$ c.$35, 41, 60, 82, 106$ d.square
Using inductive reasoning, we solve the problem in a following way: a. Triangular numbers: $1, 3, 6, 10, 15, 21$ $1 + 2 = 3$ $3+3=6$ $6+4=10$ $10+5=15$ $15+6=21$ The to get the next triangular number, we take the previous triangular number, and add one more than we did the last time. The next 5 triangular numbers are: $21+7=28$ $28+8=36$ $36+9 = 45$ $45 + 10 = 55$ $55 + 11 = 66$ b. Square numbers: $1, 4, 9, 16, 25$ $1 + 3 = 4$ $4 + 5 = 9$ $9 + 7 =16$ $16 + 9=25$ The to get the next square number, we take the previous square number, and add 2 more than we did the last time. The next 5 triangular numbers are: $25 + 11=36$ $36+13=49$ $49+15=64$ $64+17=81$ $81+19=100$ c. Pentagonal numbers: $1, 4, 9, 16, 25$ $1 + 4 = 5$ $5 + 7 = 12$ $12 + 10=22$ The to get the next pentagonal number, we take the previous pentagonal number, and add 3 more than we did the last time. The next 5 pentagonal numbers are: $22 + 13=35$ $35+16=41$ $41+19=60$ $60+22=82$ $81+25=106$ d. $1*8+1=9$ $3*8+1=25$ $6*8+1=49$ $10*8+1=81$ If a triangular number is multiplied by 8 and then 1 is added to the product, a square number is obtained.