Question

Sum of 1+3+5+7+...+997

Answer 1

1 + 3 + 5 + 7 + ... + 997

1, 3, 7, ... , 997 are terms of an arithmetic sequence

$a_1=1;\ a_2=3;\ ...;\ a_n=997\\\\d=a_2-a_1\to d=3-1=2\\\\a_n=a_1+(n-1)d\\\\subtitute\\\\997=1+(n-1)\cdot2\\977=1+2n-2\\977=2n-1\ \ \ \ |add\ 1\ to\ both\ sides\\2n=998\ \ \ \ |divide\ both\ sides\ by\ 2\\n=499\\\\S_n=\dfrac{a_1+a_n}{2}\cdot n\\\\subtitute:\ a_1=1;\ a_n=997;\ n=499\\\\S_{499}=\dfrac{1+997}{2}\cdot499=\dfrac{998}{2}\cdot499=499\cdot499=249,001$

Answer: 1 + 3 + 5 + 7 + ... + 997 = 249,001.