Master the difference between sequences and series in CBSE Class 11 Applied Mathematics. Learn definitions, formulas, and solved exercises.
A sequence is an ordered list of numbers following a specific pattern or rule, denoted as a₁, a₂, a₃, ..., aₙ. Each number in the sequence is called a term, and the position of the term is indicated by its subscript. For example, the sequence of even numbers is 2, 4, 6, 8, ... where aₙ = 2n.
Step 1: Identify the formula aₙ = 3n + 2.
Step 2: Substitute n = 5 into the formula.
Step 3: Calculate a₅ = 3(5) + 2 = 15 + 2 = 17.
Answer: 17
A series is the sum of the terms of a sequence, represented as Sₙ = a₁ + a₂ + a₃ + ... + aₙ. While a sequence is an ordered collection, a series is the result of adding those elements together. If the sequence is finite, the series is finite; otherwise, it is an infinite series.
Step 1: List the first 3 terms: a₁ = 2¹ = 2, a₂ = 2² = 4, a₃ = 2³ = 8.
Step 2: Form the series: S₃ = 2 + 4 + 8.
Step 3: Calculate the sum: 2 + 4 + 8 = 14.
Answer: 14
The fundamental difference lies in the operation: a sequence is an ordered set of numbers, whereas a series is the summation of those numbers. A sequence is a mapping from natural numbers to real numbers, while a series is a single numerical value (or a limit) representing the total sum of the sequence terms.
Step 1: Sequence for n=3 is {5, 5, 5}.
Step 2: Series for n=3 is 5 + 5 + 5 = 15.
Step 3: Note that the sequence is a list, while the series is a sum.
Answer: Sequence is {5, 5, 5}, Series is 15