General Idea:
Arithmetic sequence is a sequence which have common difference. The [tex] n^{th} [/tex] term of arithmetic sequence is given by the below formula:
[tex] a_n=a_1+(n-1)\cdot d\\ \\ Where:\\ a_1 \; is \; the \; first\; term\; of\; the\; sequence\\ n is \; the\; number \; of\; terms\; in\; the\; sequence\\ d\; is\; the\; common\; difference\; =\; a_2-a_1 [/tex]
Applying the concept:
47,42,37,32, …
[tex] a_1=47\\ \\ a_2=42\\ \\ d=a_2-a_1=42-47=-5\\ \\ a_n=47+(n-1) \cdot -5\\ a_n=47-5(n-1)\\ Distributing \; 5\; inside \; the \; parenthesis\\ \\ a_n=47-5n+5\\ Combine\; like\; terms\\ \\ a_n=52-5n [/tex]
Conclusion:
The explicit rule for the sequence 47,42,37,32 is [tex] a_n=52-5n [/tex]
