Embibe Experts Solutions for Chapter: Sequences and Series, Exercise 1: Exercise 1

Consider the sequence of numbers: $4, 7, 1, 8, 9, 7, 6,$.... For $n>2,$ the $n^{\text{th}}$ term of the sequence is the units digit of the sum of the two previous terms. Let $S_n$ denote the sum of the first $n$ terms of this sequence. What is the smallest value of $n$ for which $S_n>10,000$?

In an equilateral triangle of side $3 \mathrm{cm}$, a circle is inscribed in which again an equilateral triangle is inscribed and so on. This continues for an infinite number of times. Then

If $a_1,a_2,\dots \dots \dots a_n\in R^{+}$and $a_1·a_2\dots \dots a_n=1$ then the least value of $\left(1+a_1+a_1^2\right)\left(1+a_2+a_2^2\right)\dots .\left(1+a_n+a_n^2\right)$ is -

How many four-digit integers $abcd,$ with $a\ne 0,$ have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692,$ where $a=4, b=6, c=9,$ and $d=2.$

Let the value of $x+y+z$ is $15$, if $a, x, y, z, b$ are in A.P. while the value of $;(1 / x)+(1 / y)+(1 / z)$ is $5 / 3$ if $a, x, y, z, b$ are in H.P, then value of $\left(a^2+b^2\right)$ is

Equation $x^4+a{x}^3+b{x}^2+cx+1=0$ has real roots where $a, b, c$ are non-negative then $\min (a+b+c)$ is equal to -

Let $x^2-3x+k=0$ has two positive roots $k_1$ and $k_2$ then minimum value of $\frac{4}{k_1}+\frac{1}{k_2}$ is equal to

Let $x, y, z>0$ and $9x+3y+z=90$, then maximum value of ${\log }_{10}x+{\log }_{10}y+{\log }_{10}z$ is equal to