\displaystyle \textbf{1. Sequence}
\displaystyle \text{A sequence is a function whose domain is the set }N\text{ of all natural numbers or some subsets of the type }
\displaystyle \{1,2,3,\ldots,n\}.
\displaystyle \text{A sequence containing a finite number of terms is called a finite sequence.}
\displaystyle \text{A sequence is called an infinite sequence if it is not a finite sequence.}
\\

\displaystyle \textbf{2. Series}
\displaystyle \text{If }a_1,a_2,a_3,\ldots,a_n,\ldots\text{ is a sequence, then the expression }
\displaystyle a_1+a_2+a_3+\cdots+a_n+\cdots
\displaystyle \text{ is called a series.}
\displaystyle \text{A series is called a finite series if it has got finite number of terms, otherwise, it is called an}
\displaystyle \text{infinite series.}
\\

\displaystyle \textbf{3. Progression}
\displaystyle \text{Those sequences whose terms follow certain patterns are called progressions.}
\\

\displaystyle \textbf{4. Arithmetic Progression (A.P.)}
\displaystyle \text{A sequence is called an arithmetic progression if the difference of a term and the previous term is always same, i.e.}
\displaystyle a_{n+1}-a_n=\text{constant}(=d)\text{ for all }n\in N
\displaystyle \text{The constant difference }d\text{ is called the common difference.}
\\

\displaystyle \textbf{5. nth Term of an A.P.}
\displaystyle \text{A sequence is an arithmetic progression if and only if its nth term is a linear expression in }n
\displaystyle \text{and in such a case the common difference is equal to the coefficient of }n.
\\

\displaystyle \textbf{6. General Term of an A.P.}
\displaystyle \text{If }a\text{ is the first term and }d\text{ is the common difference of an A.P., then its nth term is given by}
\displaystyle a_n=a+(n-1)d
\\

\displaystyle \textbf{7. nth Term from the End}
\displaystyle \text{If an A.P. consists of }m\text{ terms, then nth term from the end is equal to }
\displaystyle (m-n+1)^{th}
\displaystyle \text{ term from the beginning.}
\\

\displaystyle \textbf{8. Selection of Terms in an A.P.}
\displaystyle \text{The following ways of selecting terms of an A.P. are generally very convenient:}
\displaystyle \begin{array}{|c|c|c|}\hline \text{Number of Terms} & \text{Terms} & \text{Common Difference}\\ \hline 3 & a-d,\;a,\;a+d & d\\ \hline 4 & a-3d,\;a-d,\;a+d,\;a+3d & 2d\\ \hline 5 & a-2d,\;a-d,\;a,\;a+d,\;a+2d & d\\ \hline 6 & a-5d,\;a-3d,\;a-d,\;a+d,\;a+3d,\;a+5d & 2d\\ \hline \end{array}
\\

\displaystyle \textbf{9. Sum of n Terms of an A.P.}
\displaystyle \text{The sum }S_n\text{ of }n\text{ terms of an A.P. with first term }a\text{ and common difference }d\text{ is given by}
\displaystyle S_n=\frac{n}{2}\left[2a+(n-1)d\right]
\displaystyle \text{or}
\displaystyle S_n=\frac{n}{2}(a+l)
\displaystyle \text{where }l\text{ = last term = }a+(n-1)d.
\\

\displaystyle \textbf{10. Finding the nth Term from the Sum}
\displaystyle \text{If the sum }S_n\text{ of }n\text{ terms of a sequence is given, then nth term }a_n\text{ of the sequence can be}
\displaystyle \text{determined by using the formula}
\displaystyle a_n=S_n-S_{n-1}
\\

\displaystyle \textbf{11. Condition for a Sequence to be an A.P. using }S_n
\displaystyle \text{A sequence is an A.P. iff the sum of its }n\text{ terms is of the form }
\displaystyle An^2+Bn
\displaystyle \text{i.e. a quadratic expression in }n\text{ and in such a case the common difference is twice the coefficient of }n^2.
\\

\displaystyle \textbf{12. Ratio of Sums of Two A.P.s}
\displaystyle \text{If the ratio of the sums of }n\text{ terms of two A.P.'s is given, then the ratio of their nth terms is}
\displaystyle \text{obtained by replacing }n\text{ by }(2n-1)\text{ in the given ratio.}
\\

\displaystyle \textbf{13. Arithmetic Mean}
\displaystyle \text{Three numbers }a,b,c\text{ are in A.P. iff }
\displaystyle 2b=a+c.
\displaystyle \text{In such a case }b\text{ is called the arithmetic mean of }a\text{ and }c.
\\

\displaystyle \textbf{14. Arithmetic Mean of Two Numbers}
\displaystyle \text{The arithmetic mean of }a\text{ and }b\text{ is}
\displaystyle \frac{a+b}{2}
\\

\displaystyle \textbf{15. Insertion of Arithmetic Means}
\displaystyle \text{If }n\text{ numbers }A_1,A_2,\ldots,A_n\text{ are inserted between two given numbers }a\text{ and }b\text{ such that}
\displaystyle a,\;A_1,\;A_2,\ldots,A_n,\;b
\displaystyle \text{ is an arithmetic progression, then }A_1,A_2,\ldots,A_n\text{ are known as }n\text{ arithmetic means}
\displaystyle \text{between }a\text{ and }b\text{ and the common difference of the A.P. is}
\displaystyle d=\frac{b-a}{n+1}
\displaystyle \text{Also,}
\displaystyle A_1+A_2+\cdots+A_n=n\left(\frac{a+b}{2}\right)
\\

\displaystyle \textbf{16. Sum of Equidistant Terms}
\displaystyle \text{In an A.P., the sum of the terms equidistant from the beginning and the end is always same}
\displaystyle \text{and is equal to the sum of first and last term.}
\\

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.