Question

The 11 th term of an arithmetic sequence is 58 and 31 term is 158.write the arithmetic sequence​

Answer 1

Given : -

The 11th term of an arithmetic sequence is 58 .

31st term of an arithmetic sequence is 158 .

Required to find : -

• Arithmetic progession ?

SoLuTiOn : -

11th term = 58

31st term = 158

We need to find the arithmetic progession ?

So,

We know that ;

The 11th term of an arithmetic sequence can be represented as " a + 10d "

The 31st term of an arithmetic sequence can be represented as " a + 30d "

This implies ;

a + 10d = 58 $\longrightarrow{\tt{ Equation - 1}}$

Consider this as equation - 1

a + 30d = 150 $\longrightarrow{\tt{ Equation - 2}}$

Consider this as equation - 2

Now,

We need to solve these 2 equations simultaneously .

Let's use Elimination method . So, that by eliminating one variable we can simplify our calculations .

Subtract equation 1 from equation 2

$\tt a + 30d = 158 \\ \tt a + 10d = 58 \\ \underline{ ( - )( - ) \: \: ( - ) \: \: \: \: } \\ \: \: \tt \: \: + 20d = 100 \\ \\ \implies \sf 20d = 100 \\ \\ \implies \sf d = \frac{100}{20} \\ \\ \implies \sf d = 5$

Substitute the value of d in Equation 1

=> a + 10d = 58

=> a + 10(5) = 58

=> a + 50 = 58

=> a = 58 - 50

=> a = 8

Hence,

• Common difference ( d ) = 5

• First term ( a ) = 8

Now,

Let's form the AP ;

$\to \mathsf{1st \: term \: = a = \boxed{ 8}} \\ \\ \to \mathsf{2nd \: term = a + d = 8 + 5 = \boxed{13}} \\ \\ \to \mathsf{3rd \: term = a + 2d = 8 + 2(5) = 8 + 10 = \boxed{18}} \\ \\ \to \mathsf{ 4th \: term = a + 3d = a + 3(5) = 8 + 15 = \boxed{23}}$

AP = 8 , 13 , 18 , 23 . . . . . . . . . . . .

Required AP !

AP = 8 , 13 , 18 , 23 , . . . . . . . .

Additional Information

Question

What is an Arithmetic progession ?

Answer

An arithmetic progession is a sequence of terms whose common difference is constant/equal .

Example :

• 2 , 4 , 6 , 8 , 10 . . . . . . .

• 1 , 3 , 5 , 7 , 9 . . . . . .

These are some examples of AP

Question

How to identify whether the given sequence is an AP or not ?

Answer

To identify whether any given sequence is an AP or not . We need to find the common difference between the terms . If the difference is constant then it is an AP .

The trick is ;

Common difference = ( 2nd term - 1st term ) = ( 3rd term - 2nd term )

Formulae related to arithmetic progession ;

To find the nth terms of the AP is

$\boxed{\tt{ {a}_{nth} = a + ( n - 1 ) d }}$

To find the sum of n terms of the AP is

$\boxed{\tt{ {S}_{nth} = \dfrac{n}{2} [ 2a + ( n - 1 ) d ] }}$