Find the 50th term of the A.P. 5, 11, 17, 23, ___?

Question

Prepp Team · Accepted Answer

Finding the 50th Term of an Arithmetic Progression

An Arithmetic Progression (A.P.) is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference, denoted by $d$.

The given Arithmetic Progression is: 5, 11, 17, 23, ...

Identifying the Key Elements of the A.P.

To find any term in an A.P., we need two things:

The first term ($a_1$).
The common difference ($d$).

From the given sequence:

The first term, $a_1 = 5$.
The common difference, $d$, can be found by subtracting any term from its succeeding term.
- $11 - 5 = 6$
- $17 - 11 = 6$
- $23 - 17 = 6$

So, the common difference $d = 6$.

Formula for the n-th Term of an A.P.

The formula to find the $n$-th term ($a_n$) of an Arithmetic Progression is:

$$a_n = a_1 + (n-1)d$$

Where:

$a_n$ is the $n$-th term we want to find.
$a_1$ is the first term.
$n$ is the position of the term in the sequence (e.g., 1st, 2nd, 50th).
$d$ is the common difference.

Calculating the 50th Term

We need to find the 50th term, so $n = 50$. We have $a_1 = 5$ and $d = 6$. Substituting these values into the formula:

$$a_{50} = a_1 + (50-1)d$$

$$a_{50} = 5 + (49) \times 6$$

First, calculate the product of 49 and 6:

$$49 \times 6 = 294$$

Now, add the first term to this result:

$$a_{50} = 5 + 294$$

$$a_{50} = 299$$

Therefore, the 50th term of the given Arithmetic Progression is 299.

Term Number ($n$)	Calculation using $a_n = a_1 + (n-1)d$	Term Value ($a_n$)
1	$5 + (1-1) \times 6 = 5 + 0 \times 6 = 5$	5
2	$5 + (2-1) \times 6 = 5 + 1 \times 6 = 11$	11
3	$5 + (3-1) \times 6 = 5 + 2 \times 6 = 17$	17
4	$5 + (4-1) \times 6 = 5 + 3 \times 6 = 23$	23
...	...	...
50	$5 + (50-1) \times 6 = 5 + 49 \times 6 = 5 + 294 = 299$	299

Revision Table: Key A.P. Concepts

Concept	Description	Formula
Arithmetic Progression (A.P.)	A sequence where the difference between consecutive terms is constant.	$a_1, a_1+d, a_1+2d, ...$
First Term	The initial term of the sequence.	$a_1$
Common Difference	The constant difference between consecutive terms.	$d = a_n - a_{n-1}$
n-th Term	The term at the $n$-th position in the sequence.	$a_n = a_1 + (n-1)d$

Additional Information on Arithmetic Progressions

Arithmetic Progressions are fundamental in understanding sequences and series. Beyond finding individual terms, you can also calculate the sum of the first $n$ terms of an A.P. The formula for the sum ($S_n$) is:

$$S_n = \frac{n}{2}(a_1 + a_n)$$

or

$$S_n = \frac{n}{2}(2a_1 + (n-1)d)$$

Understanding these formulas allows you to solve various problems involving arithmetic sequences, such as calculating the total number of items added over a period following an arithmetic pattern or determining the total distance covered if the distance increases arithmetically each step.