$A sequence x_{0}, x_{1}, x_{2}, x_{3}, . . . is defined by letting x_{0} = 5 and x_{k} = 4 + x_{k - 1} for all natural number k . Show that x_{n} = 5 + 4 n for all n \in N using mathematical induction .$
[NCERT EXEMPLAR]

Open in App

Solution

$Given : A sequence x_{0}, x_{1}, x_{2}, x_{3}, . . . is defined by letting x_{0} = 5 and x_{k} = 4 + x_{k - 1} for all natural number k . Let P (n) : x_{n} = 5 + 4 n for all n \in N . Step I : For n = 0, P (0) : x_{0} = 5 + 4 \times 0 = 5 So, it is true for n = 0 . Step II : For n = k, Let P (k) : x_{k} = 5 + 4 k be true for some k \in N . Step III : For n = k + 1, P (k + 1) : x_{k + 1} = 4 + x_{k + 1 - 1} = 4 + x_{k} = 4 + 5 + 4 k = 5 + 4 (k + 1) So, it is also true for n = k + 1 . Hence, x_{n} = 5 + 4 n for all n \in N .$

Suggest Corrections

Similar questions

A sequence (x_0, x_1, x_2, x_3, ......) is defined by letting $x_{0} = 5$ and $x_{k} = 4 + x_{k - 1}$ for all natural numbers k. Show that xn = 5 + 4n for all n $ϵ$ N using mathematical induction.

A sequence x_{1}, x_{2}, x_{3}, . . . is defined by letting x_{1} = 2 and x_{k} = \frac{x_{k - 1}}{k} for all natural numbers k, k \geq 2 . Show that x_{n} = \frac{2}{n!} for all n \in N .

[NCERT EXEMPLAR]

A sequence $x_{1}, x_{2}, x_{3}, . . . . . .$ is defined by letting $x_{1} = 2$ and $x_{k} = \frac{x_{k - 1}}{n}$ for all natural numbers k, $k \geq 2$ . Show that $x_{n} = \frac{2}{n!}$ for all $n ϵ N$ .

Q. Let

n

and

k

be positive integers such that

n \geq \frac{k (k + 1)}{2} .

The number of solution

(x_{1}, x_{2}, . ., x_{k}) \geq 1; x_{2} \geq 2, . . ., x_{k} \geq k

all integers satisfying

x_{1} + x_{2} + x_{3} + . . . + x_{k} = n

Using principle of mathematical induction, prove that \sqrt{n} < \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + . . . + \frac{1}{\sqrt{n}} for all natural numbers n \geq 2 .

[NCERT EXEMPLAR]

Join BYJU'S Learning Program

A sequence x0, x1, x2, x3, ... is defined by letting x0=5 and xk=4+xk-1 for all natural number k.Show that xn=5+4n for all n∈N using mathematical induction. [NCERT EXEMPLAR]

$A sequence x_{0}, x_{1}, x_{2}, x_{3}, . . . is defined by letting x_{0} = 5 and x_{k} = 4 + x_{k - 1} for all natural number k . Show that x_{n} = 5 + 4 n for all n \in N using mathematical induction .$
[NCERT EXEMPLAR]