Question

Write the first five terms of the following sequence and obtain the corresponding series: a1 = -1, an = a(n-1)/n, n ≥ 2

Answer 1

Correct Question

Write the first five terms of the following sequence and obtain the corresponding series: a₁ = -1,$\mathrm{a_{n}=\dfrac{a_{(n-1)}}{n}}$, n ≥ 2

Given:

First term, a₁= -1

nth term, $\mathrm{a_{n}=\dfrac{a_{(n-1)}}{n}}$

To Find:

First five terms of given sequence

Solution:

To obtain other terms of sequence, we have to put the values of n

↬ For n=2

$\longrightarrow\mathrm{a_{2}=\dfrac{a_{(2-1)}}{2}}$

$\longrightarrow\mathrm{a_{2}=\dfrac{a_{1}}{2}}$

$\longrightarrow\mathrm{a_{2}=\dfrac{-1}{2}}$

↬ For n=3

$\longrightarrow\mathrm{a_{3}=\dfrac{a_{(3-1)}}{3}}$

$\longrightarrow\mathrm{a_{3}=\dfrac{a_{2}}{3}}$

$\longrightarrow\mathrm{a_{3}=\dfrac{\frac{-1}{2}}{3}}$

$\longrightarrow\mathrm{a_{3}=\dfrac{-1}{6}}$

↬ For n=4

$\longrightarrow\mathrm{a_{4}=\dfrac{a_{(4-1)}}{4}}$

$\longrightarrow\mathrm{a_{4}=\dfrac{a_{3}}{4}}$

$\longrightarrow\mathrm{a_{4}=\dfrac{\frac{-1}{6}}{4}}$

$\longrightarrow\mathrm{a_{4}=\dfrac{-1}{24}}$

↬ For n=5

$\longrightarrow\mathrm{a_{5}=\dfrac{a_{(5-1)}}{5}}$

$\longrightarrow\mathrm{a_{5}=\dfrac{a_{4}}{5}}$

$\longrightarrow\mathrm{a_{5}=\dfrac{\frac{-1}{24}}{5}}$

$\longrightarrow\mathrm{a_{5}=\dfrac{-1}{120}}$

Hence, the first five terms of given sequence are $-1,\dfrac{-1}{2},\dfrac{-1}{6},\dfrac{-1}{24},\dfrac{-1}{120}$.

Answer 2

AnsWer :

• a1 = -1.
• a2 = -1/2.
• a3 = -1/6.
• a4 = -1/24.
• a5 = -1/120.
• a6 = -1/720.

Solution :

We have,

$\tt \dagger \: \: \: a_n = \frac{a_n - 1}{n}$

$\tt \dagger \: \: \: a_1 = - 1.$

$\rule{80}1$

Let's try to find five sequence.

$\tt a_n = \frac{a_{n - 1}}{n}$

$\rule{90}1$

When, n = 2.

$\tt \longmapsto \frac{a_{2 - 1}}{2}$

$\tt \longmapsto \frac{a_1}{2}$

$\tt \longmapsto \frac{ - 1}{2}$

$\rule{90}1$

When, n = 3.

$\tt \longmapsto a_n = \frac{a_{n - 1}}{n}$

$\tt\longmapsto a_ 3 = \frac{a_{3 - 1}}{3}$

$\tt\longmapsto a_3 = \frac{a_{2}}{3}$

$\tt\longmapsto a_3 = \frac{ - 1}{6}$

$\rule{90}1$

When, n = 4.

$\tt\longmapsto a_n = \frac{a_{n - 1}}{n}$

$\tt\longmapsto a_4 = \frac{a_{4 - 1}}{4}$

$\tt\longmapsto a_4 = \frac{a_3}{4}$

$\tt\longmapsto a_4 = \frac{ - 1}{24}$

$\rule{90}1$

When, n = 5.

$\tt\longmapsto a_n = \frac{a_{n - 1}}{n}$

$\tt\longmapsto a_5 = \frac{a_{5 - 1}}{5}$

$\tt\longmapsto a_5 = \frac{a_{4}}{5}$

$\tt\longmapsto a_5 = \frac{ - 1}{120}$

$\rule{90}1$

When, n = 6.

$\tt\longmapsto a_6 = \frac{a_{6 - 1}}{6}$

$\tt\longmapsto a_6 = \frac{a_{5}}{6}$

$\tt\longmapsto a_6 = \frac{ - 1}{720}$

$\rule{120}3$

Therefore, the first five terms be given sequence -1 ,-1/2 , -1/6 , -1/24 , -1/120 and -1/720.