Question

Find the common difference of an ap whose first term is 12 and 8th term is 176. Also write its 4th term.​

Answer 1

Correct Question : --

Find the common difference of an ap whose first term is 12 and 8th term is 180. Also write its 4th term.

Answer

Given : --

1st term = 12

8th term = 176

Required to find : --

• 4th term of the AP ?

Formula used : --

Formula which we need to use to find the nth term of any given arithmetic progression/sequence is ;

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

Solution : --

1st term = 12

8th term = 176

we need to find the 4th term of the AP .

So,

1st term = 12

But ,

1st term can be represented as ;

a = 12 $\longrightarrow{\bf{ Equation - 1 }}$

Consider this as Equation - 1

Similarly,

8th term = 176

But,

8th term can be represented as ;

a + 7d = 180 $\longrightarrow{\bf{ Equation - 2 }}$

Consider this as equation - 2

According to problem ;

Consider equation - 2

a + 7d = 180

Substitute the value of a from Equation -1

12 + 7d = 180

7d = 180 - 12

7d = 168

d = 168/7

d = 24

Hence,

• Common difference ( d ) = 24

Now,

let's find the 4th term ,

Using the formula

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

Here,

a = first term

d = common difference

n = no. of terms

This implies ;

$\: : \implies \sf {a}_{nth} ={a}_{4} \\ \\ : \implies \sf {a}_{4} = 12 + ( 4 - 1 ) 24 \\ \\ : \implies \sf {a}_{4} = 12 + ( 3 ) 24 \\ \\ : \implies \sf {a}_{4} = 12 + 72 \\ \\ : \implies \sf {a}_{4} = 84$

Therefore,

4th term of the AP = 84

Additional Information : --

Formula which we need to use to find the sum of n terms of any given arithmetic progression/sequence is ;

$\pink{\underline{\boxed{\rm{\purple{ {S}_{nth} = \dfrac{n}{2} [ 2a + ( n - 1 ) d ] }}}}} \huge{ \orange{ \bigstar}}$

Answer 2

$\huge{\mathcal{\underline{\purple{Solution:-}}}}$

Given First term = 12

In an AP 8th term is given by,

$\large\rm\bold{\boxed{a_8\:=\:a+7d}}$

$\large\rm{a_8\rightarrow{\:8th\:term\:in\:AP}}$

$\large\rm{a\rightarrow{First\:term\:of\:an\:AP}}$

$\large\rm{d\rightarrow{Common\:difference\:of\:AP}}$

We are given that 8th term = 176

$\large\rm{\implies{a+7d\:=\:176}}$

$\large\rm{\implies{12+7d\:=\:176}}$

$\large\rm{\implies{7d\:=\:176-12}}$

$\large\rm{\implies{7d\:=\:164}}$

$\large\rm{\implies{d\:=\:\frac{164}{7}}}$

In an AP 4th term is given by ,

$\large\rm\bold{\boxed{a_4\:=\:a+3d}}$

$\large\rm{a_4\rightarrow{\:4th\:term\:in\:AP}}$

$\large\rm{a\rightarrow{first\:term\:of\:AP}}$

$\large\rm{d\rightarrow{common\:difference\:of\:AP}}$

$\large\rm{\implies{a_4\:=\:12+3(\frac{164}{7})}}$

$\large\rm{\implies{a_4\:=\:\frac{576}{7}}}$

∴ Common difference = 164/7

Fourth term = 576/7