How many terms are there in 20, 25, 30, . . . . . . 140?

Question

How many terms are there in 20, 25, 30, . . . . . . 140?

A. 22
B. 25
C. 23
D. 24

Core Concept

This question is testing the concept of Arithmetic Progression (AP), specifically determining the number of terms in a given sequence when the first term and the last term are provided, along with the common difference.

Arithmetic Progression (AP)

An Arithmetic Progression (AP) is a sequence of numbers where each term after the first is obtained by adding a constant difference to the previous term. The common difference is denoted by $d$, and the formula for the $n$-th term of an AP is:

$$a_n = a_1 + (n – 1) \cdot d$$

Where:

• $a_n$ is the $n$-th term of the progression,
• $a_1$ is the first term,
• $d$ is the common difference,
• $n$ is the position of the term.

We need to determine the value of $n$ that corresponds to the last term in the sequence.

Step-by-Step Solution

We are given:

• The first term $a_1 = 20$,
• The common difference $d = 5$ (since the difference between consecutive terms is $25 – 20 = 5$),
• The last term $a_n = 140$.

We need to find the number of terms $n$.

Step 1: Use the formula for the $n$-th term of an AP.

We use the formula for the $n$-th term of an AP:

$$a_n = a_1 + (n – 1) \cdot d$$

Substitute the given values $a_n = 140$, $a_1 = 20$, and $d = 5$:

$$140 = 20 + (n – 1) \cdot 5$$

Step 2: Solve for $n$.

First, subtract 20 from both sides:

$$140 – 20 = (n – 1) \cdot 5$$ $$120 = (n – 1) \cdot 5$$

Now, divide both sides by 5:

$$\frac{120}{5} = n – 1$$ $$24 = n – 1$$

Finally, add 1 to both sides:

$$n = 25$$

Thus, there are 25 terms in the sequence.

Correct Answer and Option

The number of terms in the sequence is 25.

Correct Answer: B

Answer Reasoning

The problem involves applying the formula for the $n$-th term of an arithmetic progression. By substituting the known values for the first term, common difference, and last term, we were able to calculate that $n = 25$, meaning there are 25 terms in the sequence.

Explanation of Incorrect Options

• Option A (22): This is incorrect because it underestimates the number of terms, likely due to a mistake in calculating the value of $n$.
• Option C (23): This is incorrect because it suggests an overestimate of the number of terms. The correct calculation gives 25 terms.
• Option D (24): This is incorrect because it miscalculates the term count by missing the last correct value of $n = 25$.

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Common Mistakes to Avoid

1. Misunderstanding the formula: Ensure that the formula for the $n$-th term of an AP is correctly applied and that the correct values are substituted.
2. Arithmetic errors: When solving for $n$, ensure each step, especially subtraction and division, is done carefully to avoid small mistakes in the final answer.

Simplest Way to Solve (Shortcut or Insight)

The simplest way to solve this problem is:

1. Use the formula $a_n = a_1 + (n – 1) \cdot d$.
2. Substitute the known values and solve for $n$. This method directly leads to the correct result with minimal steps.

Visual Aid Suggestion

A number line could be used to visualize the arithmetic progression. Marking each term as increments of 5, starting from 20, would show how the sequence progresses, making it easier to understand how many terms are needed to reach 140.

Practice Questions

1. The first term of an AP is 10, and the common difference is 3. How many terms are there in the sequence where the last term is 100?
   • A. 30
   • B. 31
   • C. 32
   • D. 33
   • (Answer: B)
2. The first term of an AP is 7, and the common difference is 4. How many terms are there in the sequence where the last term is 51?
   • A. 12
   • B. 13
   • C. 14
   • D. 15
   • (Answer: B)
3. The first term of an AP is 3, and the common difference is 5. How many terms are there in the sequence where the last term is 53?
   • A. 9
   • B. 10
   • C. 11
   • D. 12
   • (Answer: C)
4. The first term of an AP is 15, and the common difference is 2. How many terms are there in the sequence where the last term is 45?
   • A. 15
   • B. 16
   • C. 17
   • D. 18
   • (Answer: B)
5. The first term of an AP is 5, and the common difference is 5. How many terms are there in the sequence where the last term is 130?
   • A. 25
   • B. 26
   • C. 27
   • D. 28
   • (Answer: B)