Embibe Experts Solutions for Exercise 5: Assignment

If for the sequence $\left\{a_n\right\},a_2=3,a_4=13$ and $a_7=113$, then find $a_1$ if $a_n=a{n}^2+bn+c,n\in N$.

The sum of first three consecutive terms of an $A.P.$ is $9$ and the sum of their squares is $35$ . Find $a_n$.

If the $A.M.$ between $p^{th }$ and $q^{th }$ terms of an $A.P.$ be equal to the $A.M.$ between $r^{th }$ and $s^{th }$ terms of the $A.P.$, show that $p+q=r+s$.

If $a, b, c, d$ are in $G.P.$, show that $a+b, b+c, c+d$ are in $G.P.$

Find the sum to $n$ terms of the sequence $:x+y,x^2+xy+y^2,x^3+x^{2}y+x{y}^2+y^3,\dots$

If $S_1,S_2,S_3\dots S_p$ denote the sums of infinite geometric series whose first terms are $1,2,3\dots p$ respectively and whose common ratios are $\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots \frac{1}{(p+1)}$ respectively. Show that $S_1+S_2+S_3+\dots +S_p=\frac{p(p+3)}{2}$ .

A square is drawn by joining the midpoints of sides of a given square. A third square is drawn inside the second square in the same way and this process continues indefinitely. If the side of the first square is $16 \mathrm{cm}$, determine the sum of the areas of all the squares.

Show that the sum of the cubes of any number of consecutive positive integers is divisible by the sum of those integers.