Arithmetic Progressions Ex 5.1 Class 10 NCERT Solutions & Concepts

Core Mathematical Concepts of an AP

An Arithmetic Progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed number to the preceding term. This fixed number is called the common difference ($d$).

Let the terms of an AP be $a_1, a_2, a_3, \dots, a_n$.
The common difference is calculated as:
$d = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = a_n - a_{n-1}$.

The general representation of an Arithmetic Progression with the first term '$a

It is vital to recognize that the common difference '$d

How to Verify If a Sequence Forms an AP

1. List the Terms — Identify and write down the sequence of numbers as a_1, a_2, a_3, a_4, and so on.
2. Calculate Consecutive Differences — Find the difference between successive terms: compute d_1 = a_2 - a_1, d_2 = a_3 - a_2, and d_3 = a_4 - a_3.
3. Compare the Differences — If d_1 = d_2 = d_3, then the sequence has a constant common difference and forms an AP. If the differences are not equal, it is not an AP.

Worked Solutions from Exercise 5.1 Patterns

• Example 1: Write the first four terms of the AP when first term a = -1 and common difference d = 1/2. Step 1: First term a_1 = a = -1. Step 2: Second term a_2 = a + d = -1 + 1/2 = -1/2. Step 3: Third term a_3 = a_2 + d = -1/2 + 1/2 = 0. Step 4: Fourth term a_4 = a_3 + d = 0 + 1/2 = 1/2. * Hence, the first four terms are: -1, -1/2, 0, 1/2.
• Example 2: For the AP: 3, 1, -1, -3, ... write the first term and the common difference. Step 1: The first term (a) is the first element of the series. So, a = 3. Step 2: Subtract the first term from the second term to find d: d = a_2 - a_1 = 1 - 3 = -2. Step 3: Confirm with the third term: d = a_3 - a_2 = -1 - 1 = -2. Answer: First term a = 3, Common difference d = -2.

Board Exam Alert: Common Pitfalls in AP Ex 5.1

1. Incorrect Sign of 'd': When calculating the common difference, always compute $a_2 - a_1$. Students often make the mistake of calculating $a_1 - a_2$, which reverses the algebraic sign and leads to completely incorrect sequences.

1. Ignoring Decimal/Fraction arithmetic: In questions where $a = -1.25$ and $d = -0.25$, perform signed decimal addition carefully. Remember: $-1.25 + (-0.25) = -1.50$ (not $-1.00$).

1. Real-Life Word Problems: Read situational statements carefully. If a taxi fare starts at Rs 15 and increases by Rs 8 for each additional kilometer, the terms are $15, 23, 31, \dots$ which is an AP. Do not start with $8, 23, 31\dots$.

Practice Questions with Solutions

• Q: In which of the following situations does the list of numbers involved make an arithmetic progression? 'The cost of digging a well after every metre of digging, when it costs Rs 150 for the first metre and rises by Rs 50 for each subsequent metre.' A: Step 1: Identify the starting cost (first term, a). Here, a = Rs 150. Step 2: Note the increase for every subsequent metre. The increase is Rs 50. Step 3: Write down the terms of the sequence: Cost of digging 1 metre (a_1) = Rs 150 Cost of digging 2 metres (a_2) = 150 + 50 = Rs 200 Cost of digging 3 metres (a_3) = 200 + 50 = Rs 250 Cost of digging 4 metres (a_4) = 250 + 50 = Rs 300 Step 4: Check the differences between consecutive terms: a_2 - a_1 = 200 - 150 = 50; a_3 - a_2 = 250 - 200 = 50. Since the common difference (d = 50) remains constant, the situation forms an Arithmetic Progression. Final answer: Yes, it forms an AP: 150, 200, 250, 300...
• Q: Write the first four terms of the AP when the first term is a = -1.5 and the common difference is d = -0.5. A: Step 1: Write the first term: a_1 = -1.5. Step 2: Compute the second term: a_2 = a_1 + d = -1.5 + (-0.5) = -2.0. Step 3: Compute the third term: a_3 = a_2 + d = -2.0 + (-0.5) = -2.5. Step 4: Compute the fourth term: a_4 = a_3 + d = -2.5 + (-0.5) = -3.0. Final answer: The first four terms are -1.5, -2.0, -2.5, -3.0.
• Q: Find the first term and the common difference for the following AP: 1/5, 4/5, 7/5, 10/5, ... A: Step 1: The first term 'a' is simply the first term in the sequence. Thus, a = 1/5. Step 2: Calculate the common difference 'd' by subtracting the first term from the second term: d = a_2 - a_1 = 4/5 - 1/5 = 3/5. Step 3: Double-check using the third and second terms: d = a_3 - a_2 = 7/5 - 4/5 = 3/5. Final answer: First term a = 1/5, Common difference d = 3/5.
• Q: Show whether the following list of numbers forms an AP. If it does, find the common difference d and write three more terms: 2, 8, 18, 32, ... (all values inside square roots, i.e., sqrt(2), sqrt(8), sqrt(18), sqrt(32), ...) A: Step 1: Simplify each term in radical form: a_1 = sqrt(2) a_2 = sqrt(8) = sqrt(4 2) = 2 sqrt(2) a_3 = sqrt(18) = sqrt(9 2) = 3 sqrt(2) a_4 = sqrt(32) = sqrt(16 2) = 4 sqrt(2) Step 2: Calculate consecutive differences: a_2 - a_1 = 2sqrt(2) - sqrt(2) = sqrt(2) a_3 - a_2 = 3sqrt(2) - 2sqrt(2) = sqrt(2) a_4 - a_3 = 4sqrt(2) - 3sqrt(2) = sqrt(2) Since differences are identical, the sequence forms an AP with d = sqrt(2). Step 3: Calculate the next three terms: a_5 = a_4 + d = 4sqrt(2) + sqrt(2) = 5sqrt(2) = sqrt(50) a_6 = a_5 + d = 5sqrt(2) + sqrt(2) = 6sqrt(2) = sqrt(72) a_7 = a_6 + d = 6sqrt(2) + sqrt(2) = 7*sqrt(2) = sqrt(98) Final answer: It forms an AP with d = sqrt(2). Next three terms are sqrt(50), sqrt(72), sqrt(98).

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