SSLC Mathematics Unit 1: Arithmetic Sequences

Comprehensive vocabulary and core formulas for SSLC Mathematics Unit 1, covering arithmetic sequence notation, general terms, sums, and sequence logic.

First term

Represented by the symbol $f$, it is the starting number of an arithmetic sequence.

Common difference (d)

The fixed amount added to each term to get the next term, calculated as $d = x_n - x_{n-1}$ or using the position formula $d = \frac{x_m - x_n}{m - n}$.

Nth term ($x_n$) formula (Standard)

The formula used to find any term in an arithmetic sequence: $x = f + (n - 1)d$.

Nth term ($x_n$) formula (Simplified)

An alternative way to express the nth term of a sequence: $x = dn + (f - d)$.

Algebraic form of an Arithmetic sequence

The expression $x = an + b$, where $a$ represents the common difference and $a + b$ represents the first term.

Sum of sequence ($S_n$) using First and Last terms

The formula to find the sum of $n$ terms when the first term ($x_1$ or $f$) and last term ($x_n$ or $l$) are known: $S_n = \frac{n}{2} [x_1 + x_n]$ or $S_n = \frac{n}{2} [f + l]$.

Sum of sequence ($S_n$) using First term and Difference

The general formula to calculate the sum of the first $n$ terms: $S_n = \frac{n}{2} [2f + (n - 1)d]$.

Algebraic form of Sum

The quadratic expression for the sum of an arithmetic sequence: $S_n = \frac{d}{2} n^2 + (f - \frac{d}{2})n$.

Constant Sum Property

In an arithmetic sequence, the sum of terms equidistant from the ends is constant, for example: $x_1 + x_{10} = x_2 + x_9 = x_3 + x_8$.

Difference between any two terms

The result of subtracting any two terms in an arithmetic sequence will always be a multiple of its common difference.

Middle term property for Sum

The sum of the terms in an arithmetic sequence can be found by the calculation: $\text{Number of terms} \times \text{Middle term}$.

Method for first term selection in a range (Multiples of 7 between 100 and 500)

To find the first term, divide the lower bound (100) by the divisor (7) to find the remainder (2), then calculate: $100 - 2 + 7 = 105$.

Criterion for belonging to a sequence

A number is a term of a sequence if, when divided by the common difference, it yields the same remainder as the sequence’s terms, and its position $n$ is a whole number.