Question

Write the generalized statement for the following and find the 100th term in each. i) 6, 8, 10, 12, …. ii) 10, 17, 24, 31, …​

Answer 1

$1) \: 204 \\ 2) \: 703$

Answer 2

Step-by-step explanation:

Given:

$( i) 6, 8, 10, 12,...$

$(ii) 10, 17, 24, 31,...$

To find:

$100th$ term in the following sequence.

$( i) 6, 8, 10, 12,...$

$(ii) 10, 17, 24, 31,...$  

Solution:

$( i) 6, 8, 10, 12,...$

Here the difference between each term goes like $+2.$

Here the starting number is $6$ and from that, the numbers are increasing two by two.

$1st$ term   $=6+(0).1=6$        

$2nd$ term  $=6+(1).2= 8$

$3rd$ term  $=6+(2).2=10$

This sequence goes like this.

Here are numbers $0,1,2,...$  that represent how many times $2$ is increased.

The formula can be written as follows,

⇒ $6+(n-1)2$

Here we want to find the $100th$ term of the sequence.

∴ $n=100$

⇒ $6+(100-1)2$

⇒ $6+(99)2$

⇒ $6+108$

⇒ $204$

∴ The $100th$ term in the sequence is $204.$

$(ii) 10, 17, 24, 31,...$

Here the difference between each term goes like $+7.$

Here the starting number is $10$ and from that, the numbers are increasing seven by seven.

$1st$ term   $=10+(0).7=10$

$2nd$ term  $=10+(1).7=17$

$3rd$ term  $=10+(2).7=24$

This sequence goes like this.

Here are numbers $0,1,2,...$ that represent how many times $7$ is increased.

The formula can be written as follows,

⇒ $10+(n-1)7$

Here we want to find the $100th$ term of the sequence.

∴ $n=100$

⇒ $10+(100-1)7$

⇒ $10+(99)7$

⇒ $10+693$

⇒ $703$

∴ The $100th$ term in the sequence is $703.$

Answer:

$(i)$ The $100th$ term in the sequence $6, 8, 10, 12,...$ is $204.$

$(ii)$ The $100th$ term in the sequence $10, 17, 24, 31,...$is $703.$