1. Prove that for every n ∈ N there exists k ∈ N such that n ≤ k 2 ≤ 2n 2. In the song “The Twelve Days of Christmas,” gifts are sent on successive days according to the following pattern First day: A...

1. Prove that for every n ∈ N there exists k ∈ N such that n ≤ k²
≤ 2n

2. In the song “The Twelve Days of Christmas,” gifts are sent on successive days according to the following pattern

First day: A partridge in a pear tree.

Second day: Two turtledoves and another partridge.

Third day: Three French hens, two turtledoves, and a partridge. And so on.

For each i = 1, …, 12, let gi be the number of gifts sent on the i th day. Then g₁
= 1, and for i = 2, …, 12 we have

g_i
= g_i
− 1 + i.

Now let t_n
be the total number of gifts sent during the first n days of Christmas. Find a formula for t_n
in the form

t_n=

where a, b, c ∈
.