NIMCET Maths - Arithmetic Progression: Important Concepts, Formulas & Practice Questions

Arithmetic Progression (AP) is a sequence in which the difference between consecutive terms remains constant. Questions based on AP may ask you to find the nth term, sum of terms, number of terms, or a particular term using the information provided.

NIMCET Maths Arithmetic Progression questions require you to understand the first term, common difference, number of terms, and last term and apply the appropriate formula. The PW NIMCET learning resources cover important mathematical concepts and problem-solving approaches that can help you practise AP questions systematically.

What is Arithmetic Progression?

An Arithmetic Progression (AP) is a sequence of numbers in which the difference between every pair of consecutive terms is constant. This constant difference is called the common difference.

An AP can be represented as:

a, a + d, a + 2d, a + 3d, ...

where:

• a is the first term.

• d is the common difference.

• n is the number of terms.

• The nth term is given by a + (n − 1)d.

• The common difference d remains the same for all consecutive terms.

For example, in the sequence 5, 9, 13, 17, ..., the common difference is 4.

Important Arithmetic Progression Formulas

The standard AP formulas are used to find the nth term and the sum of the first n terms.

nth Term of an AP

The nth term of an AP is:

aₙ = a + (n − 1)d

Here:

• a = first term

• d = common difference

• n = number of the term

Sum of First n Terms

The sum of the first n terms is:

Sₙ = n/2 [2a + (n − 1)d]

Alternatively, when the last term is known:

Sₙ = n/2 (a + l)

where l is the last term.

The last term can be written as:

l = a + (n − 1)d

Special Sums in Arithmetic Progression

Certain standard sequences have direct sum formulas. These formulas can be useful when solving questions involving natural numbers, odd numbers, even numbers, squares, cubes and fourth powers.

Sum of First n Natural Numbers

1 + 2 + 3 + ... + n = n(n + 1)/2

Sum of First n Odd Numbers

1 + 3 + 5 + ... + (2n − 1) = n²

Sum of First n Even Numbers

2 + 4 + 6 + ... + 2n = n(n + 1)

Sum of Squares of First n Natural Numbers

1² + 2² + ... + n² = n(n + 1)(2n + 1)/6

Sum of Cubes of First n Natural Numbers

1³ + 2³ + ... + n³ = [n(n + 1)/2]²

Sum of Fourth Powers

1⁴ + 2⁴ + ... + n⁴ = n(n + 1)(2n + 1)(3n² + 3n − 1)/30

How to Find Terms and Sums in AP?

The information given in an AP question determines which formula should be used.

• Finding the nth term: Use aₙ = a + (n − 1)d.

• Finding the sum of n terms: Use Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2(a + l).

• Finding the nth term when the sum is given: Use aₙ = Sₙ − Sₙ₋₁.

• When the last term is given: Express it as l = a + (n − 1)d and substitute it in the sum formula.

NIMCET Arithmetic Progression Examples

Example 1: Sum of the First 30 Terms

Find the sum of the first 30 terms of the AP:

5, 9, 13, 17, ...

Here:

• a = 5

• d = 4

• n = 30

Using the sum formula:

S₃₀ = 30/2 [2 × 5 + (30 − 1) × 4]

= 15 × [10 + 116]

= 15 × 126

= 1890

Therefore, the sum is 1890.

Example 2: Three-Digit Natural Numbers Divisible by 7

Find the sum of all three-digit natural numbers divisible by 7.

The first term is 105 and the last term is 994.

Here:

• a = 105

• l = 994

• d = 7

Using:

994 = 105 + (n − 1) × 7

we get:

n = 128

Now,

S = 128/2 (105 + 994)

= 64 × 1099

= 70336

Therefore, the required sum is 70336.

Example 3: Finding the 25th Term From a Given Sum

If:

Sₙ = 3n² + 5n

find the 25th term.

Using:

aₙ = Sₙ − Sₙ₋₁

we get:

aₙ = [3n² + 5n] − [3(n − 1)² + 5(n − 1)]

= 6n + 2

For n = 25:

a₂₅ = 6 × 25 + 2 = 152

Therefore, the 25th term is 152.

Shortcuts and Tricks for AP Questions

Some standard formulas can help reduce calculation time in AP questions.

• To find the nth term from a sum formula, use aₙ = Sₙ − Sₙ₋₁.

• For the sum of the first n odd numbers, directly use n².

• For the sum of the first n even numbers, use n(n + 1).

• For the sum of cubes, use [n(n + 1)/2]².

• For numbers divisible by k within a range, identify the first and last suitable numbers, calculate n, and then apply the sum formula.

NIMCET Arithmetic Progression Practice Questions

Regular practice can help you apply AP formulas to different conditions. Try solving the following questions:

1. Find the sum of all natural numbers from 1 to 100 divisible by 2 and 5.

2. A man saves ₹200 in the first month and increases his saving by ₹50 each month. In how many months will he save ₹18,600?

3. In an AP, the 15th term and the middle term are given. Find the sum of all 15 terms.

4. Find the ratio of the 11th terms of two APs given the ratio of their sums.

How to Solve NIMCET Arithmetic Progression Questions?

A systematic approach can help avoid calculation errors while solving AP questions.

1. Identify the given values: Note the first term, common difference, number of terms, last term, sum or specific term.

2. Determine what is required: Identify whether the question asks for a term, sum, number of terms or another value.

3. Choose the correct formula: Select the AP formula according to the given information.

4. Substitute the known values: Put the given values into the selected formula.

5. Solve for the unknown: Simplify the equation and calculate the required value.

6. Check special conditions: For divisibility questions, verify the first and last terms carefully.

7. Verify the answer: In MCQs, options can also be substituted to confirm the result.

NIMCET Arithmetic Progression requires a clear understanding of AP formulas, special sums and methods for finding terms and sums. Practising different question types, including divisibility, given-sum and number-of-terms problems, can help you apply these concepts accurately. PW NIMCET provides learning resources covering mathematical concepts and problem-solving approaches for NIMCET preparation.

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