Answer
a. $28, 36, 45, 55, 66$
b.$36, 49, 64, 81, 100$
c.$35, 41, 60, 82, 106$
d.square
Work Step by Step
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.
