Question

Study the following:

13 = 1 + 35 + 7 = 1 + 3 + 57 + 9 + 11 = 1 + 3 + 5 + 79 + 11 + 13 + 15

A student wrote:

1 + 3 + 5 + ... + 365367 + 369 + ... + □

What value must replace the square for this fraction to equal 13 ?

Explain how you obtained the answer.

Answer 1

The square is 731

Explanation:

Let me try...13 , 1 + 35 + 7, 1 + 3 + 57 + 9 + 11 , 1 + 3 + 5 + 79 + 11 + 13 + 15 ...(1)

grab the main number in each sequence: 13,35,57,79,911,1113....

There is another arithmetic sequence: 1, 3, 5, 7, 9, 11 ....

a₁ = 1 aₙ = 2n - 1

365367 : 2n - 1 = 365 2n = 366 n = 183 ... 365367 is in the place 183 of the series sequence

Now let's check the number follow the underlined number in (1)

0 number follow 13, 1 number follow 35, 2 number follow 57, 3 follow 79 ....

0, 1, 2, 3, 4........ until n=183 should have 182 numbers followed 365367

Now this last sequence start with 369 (a₁) with a difference of 2 from term to term

a₁₈₂ = 2n + 367 = 2 x 182 + 367 = 731