JEE Main & Advanced

  (1) Sum of \[n\] terms : The sum of n terms of an arithmetico-geometric sequence \[a,\,(a+d)\,r,\,(a+2d)\,{{r}^{2}},\,\]\[(a+3d)\,{{r}^{3}},\,.....\] is given by  \[{{S}_{n}}=\left\{ \begin{align}& \frac{a}{1-r}+dr\frac{(1-{{r}^{n-1}})}{{{(1-r)}^{2}}}-\frac{\{a+(n-1)\,d\}{{r}^{n}}}{1-r},\text{ when }r\ne 1 \\& \frac{\text{n}}{\text{2}}[2a+(n-1)\,d],\text{ when }r=1 \\\end{align} \right.\text{ }\]   (2) Sum of infinite sequence: Let \[|r|\,<1\]. Then \[{{r}^{n}},\,{{r}^{n-1}}\to 0\] as \[n\to \infty \] and it can also be shown that \[n\,.\,{{r}^{n}}\to 0\] as \[n\to \infty \]. So, we obtain that \[{{S}_{n}}\to \frac{a}{1-r}+\frac{dr}{{{(1-r)}^{2}}}\], as \[n\to \infty \]¥.   In other words, when  \[|r|\,<1\] the sum to infinity of an arithmetico-geometric series is \[{{S}_{\infty }}=\frac{a}{1-r}+\frac{dr}{{{(1-r)}^{2}}}\].

  This method is applicable for both sum of \[n\] terms and sum of infinite number of terms.   First suppose that sum of the series is \[S,\] then multiply it by common ratio of the G.P. and subtract. In this way, we shall get a G.P., whose sum can be easily obtained.  

If the differences of the successive terms of a series are in A.P. or G.P., we can find \[{{n}^{th}}\] term of the series by the following steps :   Step I: Denote the \[{{n}^{th}}\] term by \[{{T}_{n}}\] and the sum of the series upto \[n\] terms by \[{{S}_{n}}\].   Step II: Rewrite the given series with each term shifted by one place to the right.   Step III: By subtracting the later series from the former, find \[{{T}_{n}}\].   Step IV: From \[{{T}_{n}}\], \[{{S}_{n}}\] can be found by appropriate summation.   Example : Consider the series 1+ 3 + 6 + 10 + 15 +…..to \[n\] terms. Here differences between the successive terms are \[63,\text{ }106,\text{ }1510,\text{ }\ldots \ldots .\] i.e.,  2, 3, 4, 5,…… which are in A.P. This difference could be in G.P. also. Now let us find its sum   \[S=1+3+6+10+15+.....+{{T}_{n-1}}+{{T}_{n}}\]   \[S=\,\,\,\,\,\,\,\,\,1+3+6+10+..........+{{T}_{n-1}}+{{T}_{n}}\]   Subtracting, we get more...

(1) Sum of first n natural numbers   \[=1+2+3+.......+n=\sum\limits_{r=1}^{n}{r}=\frac{n\,(n+1)}{2}\].   (2) Sum of squares of first n natural numbers   \[={{1}^{2}}+{{2}^{2}}+{{3}^{2}}+.......+{{n}^{2}}=\sum\limits_{r=1}^{n}{{{r}^{2}}}=\frac{n\,(n+1)(2n+1)}{6}\].   (3) Sum of cubes of first n natural numbers   \[={{1}^{3}}+{{2}^{3}}+{{3}^{3}}+{{4}^{3}}+.......+{{n}^{3}}=\sum\limits_{r=1}^{n}{{{r}^{3}}}={{\left[ \frac{n\,(n+1)}{2} \right]}^{2}}\].  

Let \[A,\,\,G\] and \[H\] be arithmetic, geometric and harmonic means of two numbers \[a\] and \[b\].   Then, \[A=\frac{a+b}{2},\,G=\sqrt{ab}\] and \[H=\frac{2ab}{a+b}\].   These three means possess the following properties :   (1) \[A\ge G\ge H\]   \[A=\frac{a+b}{2},\,G=\sqrt{ab}\] and \[H=\frac{2ab}{a+b}\]   \[\therefore \] \[A-G=\frac{a+b}{2}-\sqrt{ab}=\frac{{{(\sqrt{a}-\sqrt{b})}^{2}}}{2}\ge 0\]\[\Rightarrow \] \[A\ge G\]   …..(i)   \[G-H=\sqrt{ab}-\frac{2ab}{a+b}=\sqrt{ab}\left( \frac{a+b-2\sqrt{ab}}{a+b} \right)=\frac{\sqrt{ab}}{a+b}{{(\sqrt{a}-\sqrt{b})}^{2}}\ge 0\]   \[\Rightarrow \] \[G\ge H\]                                                                     …..(ii)   From (i) and (ii), we get \[A\ge G\ge H\].   Note that the equality holds only when \[a=b\].   (2) \[A,\,\,G,\,\,H\] from a G.P., i.e., \[{{G}^{2}}=AH\]   \[AH=\frac{a+b}{2}\times \frac{2ab}{a+b}=ab={{(\sqrt{ab})}^{2}}={{G}^{2}}\]. Hence, \[{{G}^{2}}=AH\]   (3) The equation having \[a\] and \[b\] as its roots is   \[{{x}^{2}}-2Ax+{{G}^{2}}=0\]   The equation having \[a\] and \[b\] its roots is   \[{{x}^{2}}-(a+b)x+ab=0\]   \[\Rightarrow \] \[{{x}^{2}}-2Ax+{{G}^{2}}=0\],     \[\left[ \because A=\frac{a+b}{2}\text{ and }G=\sqrt{ab} \right]\].   The roots \[a,\,\,\,b\] are given by \[A\pm \sqrt{{{A}^{2}}-{{G}^{2}}}\].   (4) If \[A,\,\,G,\,\,H\] re arithmetic, geometric and harmonic means between three given numbers more...

  (1) If \[A,\,\,G,\,\,\,H\] be A.M., G.M., H.M. between \[a\] and \[b,\] then   \[\frac{{{a}^{n+1}}+{{b}^{n+1}}}{{{a}^{n}}+{{b}^{n}}}=\left\{ \begin{align} & A\text{ when }n=0 \\ & G\text{ when }n=-1/2 \\ & H\text{ when }n=-1 \\\end{align} \right.\]   (2) If \[{{A}_{1}},\,{{A}_{2}}\] be two A.M.?s; \[{{G}_{1}},\,{{G}_{2}}\] be two G.M.?s and \[{{H}_{1}},\,{{H}_{2}}\] be two H.M.?s between two numbers \[a\] and \[b,\] then   \[\frac{{{G}_{1}}{{G}_{2}}}{{{H}_{1}}{{H}_{2}}}=\frac{{{A}_{1}}+{{A}_{2}}}{{{H}_{1}}+{{H}_{2}}}\]   (3) Recognization of A.P., G.P., H.P. : If \[a,\,\,b,\,\,c\] are three successive terms of a sequence.   If  \[\frac{a-b}{b-c}=\frac{a}{a}\], then \[a,\,\,b,\,\,c\] are in A.P.   If, \[\frac{a-b}{b-c}=\frac{a}{b}\], then \[a,\,\,b,\,\,c\] are in G.P.   If, \[\frac{a-b}{b-c}=\frac{a}{c}\], then \[a,\,\,b,\,\,c\] are in H.P.   (4) If number of terms of any A.P./G.P./H.P. is odd, then A.M./G.M./H.M. of first and last terms is middle term of series.   (5) If number of terms of any A.P./G.P./H.P. is even, then A.M./G.M./H.M. of middle two terms is A.M./G.M./H.M. of first and last terms respectively.   (6) If \[{{p}^{th}},\,\,{{q}^{th}}\] more...

Let us consider three homogeneous linear equations   \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z=0\],\[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z=0\]   and  \[{{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}z=0\]   Eliminated \[x,\,\,y,\,\,z\] from above three equations we obtain   \[{{a}_{1}}({{b}_{2}}{{c}_{3}}-{{b}_{3}}{{c}_{2}})-{{b}_{1}}({{a}_{2}}{{c}_{3}}-{{a}_{3}}{{c}_{2}})+{{c}_{1}}({{a}_{2}}{{b}_{3}}-{{a}_{3}}{{b}_{2}})=0\]   …..(i)   The L.H.S. of (i) is represented by  \[\left| \,\begin{matrix}{{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\{{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\{{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\\end{matrix}\, \right|={{a}_{1}}\,\left| \,\begin{matrix}{{b}_{2}} & {{c}_{2}}  \\{{b}_{3}} & {{c}_{3}}  \\\end{matrix}\, \right|-{{b}_{1}}\,\left| \,\begin{matrix}{{a}_{2}} & {{c}_{2}}  \\{{a}_{3}} & {{c}_{3}}  \\\end{matrix}\, \right|+{{c}_{1}}\,\left| \,\begin{matrix}{{a}_{2}} & {{b}_{2}}  \\{{a}_{3}} & {{b}_{3}}  \\\end{matrix}\, \right|\]   Its contains three rows and three columns, it is called a determinant of third order.   The number of elements in a second order is \[{{2}^{2}}=4\] and the number of elements in a third order determinant is \[{{3}^{2}}=9\].   Rows and columns of a determinant : In a determinant horizontal lines counting from top \[{{1}^{st}},\text{ }{{2}^{nd}},\text{ }{{3}^{rd}},\ldots ..\] respectively known as rows and denoted by \[{{R}_{1}},\,\,{{R}_{2}},\,\,{{R}_{3}},\,\,......\] and vertical lines counting left to right, \[{{1}^{st}},\text{ }{{2}^{nd}},\text{ more...

P-1 : The value of determinant remains unchanged, if the rows and the columns are interchanged.   Since the determinant remains unchanged when rows and columns are interchanged, it is obvious that any theorem which is true for ‘rows’ must also be true for ‘columns’.   P-2 : If any two rows (or columns) of a determinant be interchanged, the determinant is unaltered in numerical value but is changed in sign only.   P-3 : If a determinant has two rows (or columns) identical, then its value is zero.   P-4 : If all the elements of any row (or column) be multiplied by the same number, then the value of determinant is multiplied by that number.   P-5 : If each element of any row (or column) can be expressed as a sum of two terms, then the determinant can be expressed as the sum of the determinants.   more...

  (1) Minor of an element : If we take the element of the determinant and delete (remove) the row and column containing that element, the determinant left is called the minor of that element. It is denoted by \[{{M}_{ij}}\].   Consider the determinant \[\Delta =\left| \,\begin{matrix} {{a}_{11}} & {{a}_{12}} & {{a}_{13}}  \\ {{a}_{21}} & {{a}_{22}} & {{a}_{23}}  \\ {{a}_{31}} & {{a}_{32}} & {{a}_{33}}  \\ \end{matrix}\, \right|\],   then determinant of minors \[M=\left| \,\begin{matrix} {{M}_{11}} & {{M}_{12}} & {{M}_{13}}  \\ {{M}_{21}} & {{M}_{22}} & {{M}_{23}}  \\ {{M}_{31}} & {{M}_{32}} & {{M}_{33}}  \\ \end{matrix}\, \right|\]   where  \[{{M}_{11}}=\] minor of \[{{a}_{11}}=\left| \,\begin{matrix} {{a}_{22}} & {{a}_{23}}  \\ {{a}_{32}} & {{a}_{33}}  \\ \end{matrix}\, \right|\]  \[{{M}_{12}}=\]minor of  \[{{a}_{12}}=\left| \,\begin{matrix} {{a}_{21}} & {{a}_{23}}  \\ {{a}_{31}} & {{a}_{33}}  \\ \end{matrix}\, \right|\] \[{{M}_{13}}=\] minor of \[{{a}_{13}}=\left| \,\begin{matrix} {{a}_{21}} & {{a}_{22}}  \\  {{a}_{31}} & {{a}_{32}}  \\ \end{matrix}\, \right|\]   Similarly, we can find the minors of other elements more...

 Let the two determinants of third order be,   \[{{D}_{1}}=\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix} \right|\] and \[{{D}_{2}}=\left| \,\begin{matrix} {{\alpha }_{1}} & {{\beta }_{1}} & {{\gamma }_{1}}  \\ {{\alpha }_{2}} & {{\beta }_{2}} & {{\gamma }_{2}}  \\ {{\alpha }_{3}} & {{\beta }_{3}} & {{\gamma }_{3}}  \\ \end{matrix}\, \right|\].   Let D be their product.   Then \[D=\left| \,\begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}}  \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}}  \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}}  \\ \end{matrix} \right|\,\,\,\times \left| \,\begin{matrix} {{\alpha }_{1}} & {{\beta }_{1}} & {{\gamma }_{1}}  \\ {{\alpha }_{2}} & {{\beta }_{2}} & {{\gamma }_{2}}  \\ {{\alpha }_{3}} & {{\beta }_{3}} & {{\gamma }_{3}}  \\ \end{matrix}\, \right|\]     \[=\left| \,\begin{matrix} {{a}_{1}}{{\alpha }_{1}}+{{b}_{1}}{{\beta }_{1}}+{{c}_{1}}{{\gamma }_{1}} & {{a}_{1}}{{\alpha }_{2}}+{{b}_{1}}{{\beta }_{2}}+{{c}_{1}}{{\gamma }_{2}} & {{a}_{1}}{{\alpha }_{3}}+{{b}_{1}}{{\beta }_{3}}+{{c}_{1}}{{\gamma }_{3}}  \\ {{a}_{2}}{{\alpha }_{1}}+{{b}_{2}}{{\beta }_{1}}+{{c}_{2}}{{\gamma }_{1}} & {{a}_{2}}{{\alpha }_{2}}+{{b}_{2}}{{\beta more...