Series and Progressions

Series and Progressions

Series and progressions are important topics in quantitative aptitude exams like TCS NQT. Let's discuss some key concepts and problem-solving techniques related to series and progressions that you may encounter in the exam.

Arithmetic Progression (AP):

Definition: An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. The constant difference is called the common difference (d).

General Term: The nth term of an AP is given by the formula: a_n = a + (n - 1)d, where a is the first term and n is the position of the term.

Sum of Terms: The sum of the first n terms of an AP can be calculated using the formula: S_n = (n/2)[2a + (n-1)d].

Problems involving finding the nth term, the sum of terms, or the number of terms in an AP are common in AP problems.

Example 1: Find the sum of the first 50 terms of an arithmetic progression if the first term is 5 and the common difference is 3.

Solution: The formula to find the sum of the first n terms of an AP is: Sn = (n/2)[2a + (n-1)d]

Given:

First term (a) = 5

Common difference (d) = 3

Number of terms (n) = 50

Substituting the values into the formula:

Sn = (50/2)[2(5) + (50-1)(3)]

= 25[10 + 49(3)]

= 25[10 + 147]

= 25(157)

= 3925

Therefore, the sum of the first 50 terms of the given arithmetic progression is 3925.

Example 2: Find the 15th term of an arithmetic progression if the first term is 7 and the common difference is -2.

Solution: The formula to find the nth term of an AP is: an = a + (n-1)d

Given:

First term (a) = 7

Common difference (d) = -2

Position of the term (n) = 15

Substituting the values into the formula:

a15 = 7 + (15-1)(-2)

= 7 + 14(-2)

= 7 - 28

= -21

Therefore, the 15th term of the given arithmetic progression is -21.

These examples illustrate the application of formulas for finding the sum of terms or the nth term in an arithmetic progression. Practice more problems to strengthen your understanding and problem-solving skills in arithmetic progressions.

Geometric Progression (GP):

Definition: A geometric progression is a sequence of numbers in which the ratio between consecutive terms is constant. The constant ratio is called the common ratio (r).

General Term: The nth term of a GP is given by the formula: a_n = a * r^(n-1), where a is the first term and n is the position of the term.

Sum of Terms: The sum of the first n terms of a GP can be calculated using the formula:

If r < 1, S_n = a * (1 - r^n) / (1 - r).

If r > 1, S_n = a * (r^n - 1) / (r - 1).

Problems involving finding the nth term, the sum of terms, or the number of terms in a GP are common in GP problems.

Examples:

Example 1: Find the sum of the first 6 terms of a geometric progression if the first term is 3 and the common ratio is 2.

Solution: The formula to find the sum of the first n terms of a GP is: Sn = a * (1 - r^n) / (1 - r)

Given:

First term (a) = 3

Common ratio (r) = 2

Number of terms (n) = 6

Substituting the values into the formula:

Sn = 3 * (1 - 2^6) / (1 - 2)

= 3 * (1 - 64) / (1 - 2)

= 3 * (-63) / (-1)

= 189

Therefore, the sum of the first 6 terms of the given geometric progression is 189.

Example 2: Find the 10th term of a geometric progression if the first term is 2 and the common ratio is 0.5.

Solution: The formula to find the nth term of a GP is: an = a * r^(n-1)

Given:

First term (a) = 2

Common ratio (r) = 0.5

Position of the term (n) = 10

Substituting the values into the formula:

a10 = 2 * (0.5)^(10-1)

= 2 * (0.5)^9

= 2 * (1/2)^9

= 2 * (1/512)

= 1/256

Therefore, the 10th term of the given geometric progression is 1/256.

These examples illustrate the application of formulas for finding the sum of terms or the nth term in a geometric progression. Practice more problems to strengthen your understanding and problem-solving skills in geometric progressions.

Harmonic Progression (HP):

Definition: A harmonic progression is a sequence of numbers in which the reciprocals of the terms form an arithmetic progression.

General Term: The nth term of an HP is given by the formula: a_n = 1 / (a + (n - 1)d), where a is the first term and d is the common difference.

Sum of Terms: The sum of the first n terms of an HP can be calculated using the formula: S_n = n / [(2a + (n - 1)d) * n].

Problems involving finding the nth term, the sum of terms, or the number of terms in an HP are common in HP problems.

Example 1: Find the sum of the first 5 terms of a harmonic progression if the first term is 1 and the common difference is 1.

Solution: The formula to find the sum of the first n terms of an HP is: Sn = n / [(2a + (n-1)d) * n]

Given:

First term (a) = 1

Common difference (d) = 1

Number of terms (n) = 5

Substituting the values into the formula:

Sn = 5 / [(2(1) + (5-1)(1)) * 5]

= 5 / [(2 + 4) * 5]

= 5 / (6 * 5)

= 5 / 30

= 1/6

Therefore, the sum of the first 5 terms of the given harmonic progression is 1/6.

Example 2: Find the 8th term of a harmonic progression if the first term is 1 and the common difference is 1/2.

Solution: The formula to find the nth term of an HP is: an = 1 / (a + (n-1)d)

Given:

First term (a) = 1

Common difference (d) = 1/2

Position of the term (n) = 8

Substituting the values into the formula:

a8 = 1 / (1 + (8-1)(1/2))

= 1 / (1 + 7/2)

= 1 / (9/2)

= 2/9

Therefore, the 8th term of the given harmonic progression is 2/9.

These examples illustrate the application of formulas for finding the sum of terms or the nth term in a harmonic progression. Practice more problems to strengthen your understanding and problem-solving skills in harmonic progressions.

Tips for Problem Solving:

Identify the type of progression (AP, GP, or HP) by observing the pattern in the given series.

Write down the given terms and try to find the common difference or common ratio.

Use the appropriate formulas to find the missing terms, sum of terms, or number of terms.

Practice solving different types of problems to improve your speed and accuracy.