Question

Computer found that there are 1000 constellations in the sky. The first of them consists of one star, the second of them consists of three stars, the third - five stars and so on (in each of the next constellations there are two stars more than in the previous one). In the last constellation there are 1999 stars. How many stars in the sky in total?

A. 999,000
B. 2,000,000
C. 1,000,000
D. 1,999,000

Answer 1

The total number of stars in the 1000 constellations is 1,000,000, determined by using the formula for the sum of the first n terms of an arithmetic sequence. Therefore, the correct answer is option C.

Step-by-step explanation:

The student's question is about finding the total number of stars in the sky when given a sequence defining the number of stars in each of 1000 constellations, with the number of stars in each constellation following a predictable pattern. Specifically, the first constellation contains one star, and each subsequent constellation contains two more stars than the previous one, up to the last which contains 1999 stars.

This is a mathematical sequence known as an arithmetic sequence, where the nth term is given by the equation Tn = a + (n-1)d, where 'a' is the first term, 'd' is the common difference and 'n' the term number. In this case, 'a' is 1, 'd' is 2, and 'n' is 1000. The sum 'S' of the first 'n' terms is given by the equation S = n/2 * (a + Tn). By substituting the given values, we calculate the total number of stars in the constellations as follows:

Sum of stars = 1000/2 * (1 + 1999) = 500 * (2000) = 1,000,000.

Therefore, the total number of stars in the sky, according to the given pattern, is 1,000,000, which corresponds to option C.