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Applications Of The Nth Term

Use the nth term formula to find 'n', check if a number is in a sequence, and solve word problems.

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Using the Formula in Reverse

Explain how to solve for 'n' when the term value is known.

Finding the Position of a Term

We know that the formula $a_n = a + (n - 1)d$ is used to find the value of the $n$ th term. But we can also use it in reverse! If we know the value of a specific term ( $a_n$ ), we can work backward to find its position ( $n$ ) in the sequence.

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Applications of the nth Term

Shows a real-world scenario like the wood logs

$A photorealistic educational scene of a beautifully arranged flower bed with rows of rose plants steadily decreasing in number from the front row to the back row, shot on 35mm lens, natural lighting with soft bokeh, warm inviting color grading, golden hour or soft diffused daylight, professional documentary photography quality.$

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The nth term formula helps calculate exactly how many rows fit in a space when each row decreases by a set amount.

Checking if a Number is in an AP

Example 6: Checking if 301 is a term.

Problem. Check whether $301$ is a term of the list of numbers $5, 11, 17, 23, \dots$

Here, $a = 5$ and $d = 11 - 5 = 6$ .
Let $301$ be a term, say, the $n$ th term of this AP. So, $a_n = 301$ .

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Finding How Many Terms Satisfy a Condition

Example 7: Two-digit numbers divisible by 3.

Problem. How many two-digit numbers are divisible by 3?

The list of two-digit numbers divisible by $3$ is: $12, 15, 18, \dots, 99$ .
This forms an AP where $a = 12$ , $d = 3$ , and the last term $a_n = 99$ .

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Finding a Term from the End

Example 8: Finding 11th term from the last term.

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To find the 11th term from the last term of the AP $10, 7, 4, \dots, -62$ , we first find the total number of terms.

Using $-62 = 10 + (n - 1)(-3)$ , we solve to find $n - 1 = 24$ , giving us a total of terms in the AP.

The 11th term from the last term will be the th term from the beginning.

We calculate this term as $a_{15} = 10 + (15 - 1)(-3) = 10 - 42$ , which equals .

Interpreting 'n'

Test understanding of what n represents.

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While checking if $-150$ is a term of the AP $11, 8, 5, 2\dots$ , a student sets $a_n = -150$ and finds $n = 164/3$ . What is the correct mathematical conclusion?

Rows in a Flower Bed

Solve a real-world AP problem.

Solve the problem below step-by-step.

Problem: In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so on. There are 5 rose plants in the last row. How many rows are there in the flower bed?

Known Values

Identify the first term, common difference, and the nth term.

Formula

State the formula you will use.

Substitution

Substitute your known values into the formula.

Calculation Steps

Show the steps to isolate and solve for n.

Final Answer

State the final number of rows.

Check (Optional)

Does your answer make sense for a physical number of rows?

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Using the Formula in Reverse

Explain how to solve for 'n' when the term value is known.

Finding the Position of a Term

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Checking if a Number is in an AP

Example 6: Checking if 301 is a term.

Problem. Check whether $301$ is a term of the list of numbers $5, 11, 17, 23, \dots$

Here, $a = 5$ and $d = 11 - 5 = 6$ .
Let $301$ be a term, say, the $n$ th term of this AP. So, $a_n = 301$ .

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Finding How Many Terms Satisfy a Condition

Example 7: Two-digit numbers divisible by 3.

Problem. How many two-digit numbers are divisible by 3?

The list of two-digit numbers divisible by $3$ is: $12, 15, 18, \dots, 99$ .
This forms an AP where $a = 12$ , $d = 3$ , and the last term $a_n = 99$ .

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Finding a Term from the End

Example 8: Finding 11th term from the last term.

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To find the 11th term from the last term of the AP $10, 7, 4, \dots, -62$ , we first find the total number of terms.

Using $-62 = 10 + (n - 1)(-3)$ , we solve to find $n - 1 = 24$ , giving us a total of terms in the AP.

The 11th term from the last term will be the th term from the beginning.

We calculate this term as $a_{15} = 10 + (15 - 1)(-3) = 10 - 42$ , which equals .

Interpreting 'n'

Test understanding of what n represents.

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While checking if $-150$ is a term of the AP $11, 8, 5, 2\dots$ , a student sets $a_n = -150$ and finds $n = 164/3$ . What is the correct mathematical conclusion?