The image contains several math problems. I will solve "Example 1" and "Related Problem 1". Example 1: A sequence is formed by adding consecutive integers to 1. The sequence is 1, 2, 4, 7, 11, 16, ... What is the 15th number in the sequence? Related Problem 1: In the sequence from Example 1, what is the 20th number?

1. Problem Description

The image contains several math problems. I will solve "Example 1" and "Related Problem 1".

Example 1: A sequence is formed by adding consecutive integers to

1. The sequence is 1, 2, 4, 7, 11, 16, ... What is the 15th number in the sequence?

Related Problem 1: In the sequence from Example 1, what is the 20th number?

2. Solution Steps

Example 1:

The sequence is formed by adding consecutive integers starting from 1 to the previous term.

1st term: 1

2nd term: 1 + 1 = 2

3rd term: 2 + 2 = 4

4th term: 4 + 3 = 7

5th term: 7 + 4 = 11

...

nth term is given by: $1 + (1 + 2 + 3 + ... + (n-1))$.

The sum of the first $n-1$ integers is given by the formula:

$S = \frac{(n-1)n}{2}$

Therefore, the nth term is $1 + \frac{(n-1)n}{2}$.

For the 15th term ($n=15$), the number is:

$1 + \frac{(15-1) \times 15}{2} = 1 + \frac{14 \times 15}{2} = 1 + 7 \times 15 = 1 + 105 = 106$

Related Problem 1:

To find the 20th number in the sequence, we substitute $n = 20$ into the formula:

$1 + \frac{(20-1) \times 20}{2} = 1 + \frac{19 \times 20}{2} = 1 + 19 \times 10 = 1 + 190 = 191$

3. Final Answer

Example 1: The 15th number is

1

0

6. Related Problem 1: The 20th number is

1

9

1.