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.

4 0

3 years ago

Suppose at one point along the Nile River a ferryboat must travel straight across a 1.40-mile stretch from west to east. At this

pogonyaev

Answer:

about 14° south of east

Step-by-step explanation:

The river speed is about ...

(2.35 m/s)×(1 mi/(1609.344 m))×(3600 s)/(1 h) ≈ 5.2568 mi/h

The angle of interest is such that its sine is the ratio of river speed to boat speed:

sin(α) = (5.2568 mi/h)/(21.8 mi/h) = 0.241138

α = arcsin(0.241138) = 13.9537°

Since the river is flowing north, the boat should be directed south of its intended course by this angle.

The boat should be directed 14° south of east.

7 0

4 years ago