Question

Complete the table and pattern. Determine the common difference. Write the explicit rule.

Answer 1

6.

$a_1=529,\ a_2=508\\\\d=a_2-a_1\to d=508-529=-21\\\\a_n=a_1+(n-1)d\to a_n=529+(n-1)(-21)=529-21n+21=550-21n\\\\a_4=550-21(4)=550-84=466\\\\a_5=550-21(5)=445\\\\a_6=550-21(6)=424$

n   |    1     |   2    |   3   |   4   |   5    |   6   |

f(n)|  529 | 508 | 487 | 466 | 445 | 424 |

First term: 529      Common Difference: -21

Explicit Rule: $a_n=550-21n$       Recursive Rule: $a_1=529,\ a_n=a_{n-1}-21$

7.

$a_1=7.5,\ a_2=9\\\\d=a_2-a_1\to d=9-7.5=1.5\\\\a_n=a_1+(n-1)d\to a_n=7.5+(n-1)(1.5)=7.5+1.5n-1.5=1.5n+6\\\\a_4=1.5(4)+6=6+6=12\\\\a_6=1.5(6)+6=9+6=15$

n   |    1     |   2    |   3   |   4   |   5    |   6   |

f(n)|  7.5   |   9    |10.5 |  12  |  13.5 |   15   |

First term: 7.5      Common Difference: 1.5

Explicit Rule: $a_n=1.5n+6$       Recursive Rule: $a_1=7.5,\ a_n=a_{n-1}+1.5$

8.

$a_2=120,\ a_3=145\\\\d=a_3-a_2\to d=145-120=25\\\\\a_1=a_2-d\to a_1=120-25=95\\\\a_n=a_1+(n-1)d\to a_n=95+(n-1)(25)=95+25n-25=25n+70\\\\a_5=25(5)+70=125+70=195\\\\a_6=25(6)+70=150+70=220$

n   |   1    |   2    |   3   |   4   |   5    |   6   |

f(n)|  95  | 120  | 145 |  170 | 195 | 220 |

First term: 95      Common Difference: 25

Explicit Rule: $a_n=25n+70$       Recursive Rule: $a_1=95,\ a_n=a_{n-1}+25$