What is the Next Number? 0, 3, 8, 15, 24, 35

Numbers 0, 3, 8, 15, 24, and 35 are not forming any sequence. The difference between two consecutive numbers in the given sequence forms a new sequence. This sequence where the difference between two numbers situated next to each other remains constant is called Arithmetic Progression (AP).

• When we subtract the first two numbers of the sequence that is 0 and 3, we get the result as 3
• When we subtract the next two numbers of the sequence that is 3 and 8, we get the result as 5
• When we subtract the next two numbers of the sequence that is 8 and 15, we get the result as 7
• When we subtract the next two numbers of the sequence that is 15 and 24, we get the result as 9
• When we subtract the next two numbers of the sequence that is 24 and 35, we get the result 11.

So in this way the next number in the sequence, 0, 3, 8, 15, 24, 35, is 48.

As the next term in the new sequence will be 13.

So, 35 + 13 = 48.

New obtained sequence is 3, 5, 7, 9, 11, 13, and so on.

This sequence is in arithmetic progression. Having common difference (d) as 2, the first term (a) as 3.

The general expression of an arithmetic progression is given below:

an = a + (n - 1) x d

where an is the general term.

a is the first term of the sequence.

d is a common difference.

n is the position of any random term in the sequence.

The formula to calculate the common difference (d) of any sequence is:

(b - a) = (c - b) = d

Or b + b = a + c

Or 2 x b = (a + c)

Or b = (a + c) / 2

Some examples of sequences that are in arithmetic progression are

1, 2, 3, 4, 5, .... (First term (a) is 1, and common difference (d) is 1)

0, 5, 10, 15, 20, .... (First term (a) is 0, and common difference (d) is 5)

2, 4, 6, 8, 10, .... (First term (a) is 2, and common difference (d) is 2)

10, 20, 30, 40, 50, .... (First term (a) is 10, and common difference (d) is 10)

Arithmetic sequences have many applications in various fields, such as mathematical modeling, physics, and finance. For example, in physics, an arithmetic sequence can be used to model the displacement of a moving object. They also play a key role in solving mathematical problems.

In conclusion, an arithmetic sequence is a set of numbers where the difference between any two terms beside each other remains constant. They are frequently utilized in many different sectors and are essential to solving mathematical issues. The arithmetic series, the sum of an arithmetic sequence's terms, is a special case of an arithmetic sequence denoted by Sn.

Sn = (n / 2) (2a + (n - 1) x d)

Where:

Sn denotes the sum of the sequence => series.

n is the total number of terms in the sequence.

a is the first term.

d is a common difference.

Few Examples of Mathematical Problems that Involve Arithmetic Sequences

1. Find the 10th term of the arithmetic sequence: 2, 5, 8, 11, 14, ...

Solution: To find the 10th term of this sequence, we can use the formula for the nth term of an arithmetic sequence: an = a1 + (n-1) d

In this case, a1 = 2, d = 3, and n = 10.

Substituting these values into the formula, we get:

= a10 = 2 + (10JGJ-1)3

= 2 + 9*3

= 2 + 27 = 29

So, the 10th term of the sequence is 29.

2. Find the common difference of the arithmetic sequence: -5, -1, 3, 7, 11, ...

Solution: To find the common difference in the given sequence, we can find the difference between two neighboring terms like -5 and -1 or 7 and 11, etc. In this case, we can subtract -5 and -1 to get 4, subtract -1 and 3 to get 4, subtract 3 and 7 to get 4, and then subtract 7 and 11 to get 4.

Since the result is always 4,

therefore, the common difference of this sequence is 4.

3. Find the sum of the first 20 terms of the arithmetic sequence: 5, 10, 15, 20, 25,...

Solution: To find the sum of the first 20 terms of this sequence, we can use the formula for the sum of the first n terms of an arithmetic series:

Sn = (n / 2) (2a + (n - 1) x d)

In this case, a = 5, n = 20 and d = 5

Sn = (20/2) (2*5+(20-1)5

Sn = 10*105

Sn = 1050

So, the sum of the first 20 terms of the sequence is 1050.