Question

Question 12 Write the first five terms of the following sequence and obtain the corresponding series: a1 = -1, an = a(n-1)/n, n ≥ 2

Class X1 - Maths -Sequences and Series Page 181

Answer 1

$a_1=-1,a_n=\frac{a_{n-1}}{n}, where, n \geq 2\\\\\\a_2=\frac{a_{2-1}}{2}\\=\frac{a_1}{2}\\=\frac{-1}{2}\\\\a_3=\frac{a_{3-1}}{3}\\=\frac{a_2}{3}\\=\frac{\frac{-1}{2}}{3}\\=\frac{-1}{6}\\\\a_4=\frac{a_{4-1}}{4}\\=\frac{a_3}{4}\\=\frac{\frac{-1}{6}}{4}\\=\frac{-1}{24}\\\\a_5=\frac{a_4}{5}\\=\frac{\frac{-1}{24}}{5}\\=\frac{-1}{120}\\\\a_6=\frac{a_5}{6}\\=\frac{\frac{-1}{120}}{6}\\=\frac{-1}{720}$

hence, first five terms are -1/2 , -1/6 , -1/24 , -1/120, -1/720

Answer 2

Given:

a1 = - 1

an = a(n-1)/n, n ≥ 2

To Find:

The First five terms of the following sequence and obtain the corresponding series.

Calculating:

When we put the value of n as 2:

a2 = a2 - 1 / 2

= a1 / 2  

= - 1 / 2 (Putting Value of a1 = - 1)

When we put the value of n as 3:

a3 = a3 - 1 / 3

= a2 / 3

= - 1/2 / 3 (Putting Value of a2 = -1/2)

Rearranging to multiply:

= (- 1/2) (1/3)

= - 1/6

When we put the value of n as 4:

a4 = a4-1 / 4

= a3 / 4

= -1/6 / 4 (Putting the value of a3 = -1/6)

Rearranging to multiply:

= (-1/6)(1/4)    

= -1/24

When we put the value of n as 5:

a5 = a5-1 / 5

= a4 / 5

= -1/24 / 5 (Putting the value of a4 -1/24)

Rearranging to multiply:

= (- 1/24) (1/5)

= -1/120

Therefore, the first five terms of this sequence is -1, -1/2, -1/6, -1/24,-1/20.

Obtaining the corresponding series (-1)+ (-1/2) + (-1/6) + (-1/24) + (-1/120)+.......