Class-10 Mathematics-Chapter-1 Arithmetic Sequence Important points
- A set of numbers written as the first, second, third and so on, is called a sequence.
- The numbers forming a sequence are called its terms.
- A sequence got by starting with any number and adding a fixed number repeatedly is called an arithmetic sequence.
- An arithmetic sequence in which we get the same number on subtracting from any term, the term immediately preceding it.
- This constant difference got by subtracting from any term the just previous term, is called the common difference of an arithmetic sequence.
- The difference between any two terms of an arithmetic sequence is the product of the difference of positions and the common difference.
- In an arithmetic sequence, term difference is proportional to position difference; and the constant of proportionality is the common difference.
- In any arithmetic sequence of natural numbers, the difference of any two terms is a multiple of the common difference. This means they leave the same remainder on division by the common difference.
- Any arithmetic sequence of natural numbers is of the form described first; it consists of numbers leaving the same remainder on division by a specific number. And this divisor is the common difference.
- In an arithmetic sequence, the sum of the first five times the middle term and the sum first seven terms is seven times the middle term.
- Any arithmetic sequence is of the form xn = an + b where a and b are fixed numbers; conversely, any sequence of this form is an arithmetic sequence.
- The sum of any number of consecutive natural numbers, starting with one, is half the product the last number and the next natural number.
- For the arithmetic sequence
xn = an + b
the sum of the first n terms is
x1, + x2 + ... + xn = ½ an(n + 1) + nb - The sum of any number of consecutive terms of an arithmetic sequence is half the product of the number of terms and the sum of the first and last terms.