Question

Write the arithmetic sequence with first term 30 and sum of six term is 300

Answer 1

Answer:

answer for the given problem is given

Answer 2

Given :-

First term = 30

Sum of six term is 300

Required to find :-

• Arithmetic progession/sequence ?

Solution :-

Given data :-

First term is 30

Sum of six term is 300

we need to find the arithmetic sequence .

So,

From the given information let's find the value of d .

Since,

First term ( a ) = 30

And,

Sum of first 6 terms = 300

This is also represented as ;

$\tt{ {S}_{nth} = {S}_{6} }$

$\tt{ {S}_{6} = \dfrac{ 6}{2} [ 2 a + ( 6 - 1 )d ] }$

$\tt{ {S}_{6} = \dfrac{ 6}{2} [ 2 ( 30 ) + ( 5 ) d ]}$

$\tt{ \because {S}_{6} = 300 }$

$\tt{ 300 = \dfrac{6}{ 2 } [ 60 + 5d ] }$

$\tt{ 300 = 3 \times ( 60 + 5d ) }$

$\tt{ 300 = 180 + 15d }$

$\tt{ 300 - 180 = 15d }$

$\tt{ 120 = 15d }$

$\tt{ 15d = 120 }$

$\tt{ d = \dfrac{120}{15} }$

$\tt{ d = 8 }$

Hence,

• Common difference ( d ) = 8

Now,

Let's find the arithmetic sequence ;

$\tt{1st \: term = a = \boxed{30}} \\ \\ \tt{2nd \: term = a + d = 30 + 8 = \boxed{38}} \\ \\ \tt{3rd \: term = a + 2d = 30 + 2 \times 8 = 30 + 16 = \boxed{46}} \\ \\ \tt{4th \: term = a + 3d = 30 + 3 \times 8 = 30 + 24 = \boxed{54}}$

Therefore,

The required arithmetic sequence is ;

$\tt{ \red{ AP} \blue{ =} \green{ 30 \: , \: 38 \:, \: 46 \: , \: 54 \: , \dots \dots \dots }}$

Additional Information :-

To find the nth term of any given AP . The formula is ;

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

Similarly,

To find the sum of nth terms of any given AP. The formula is ;

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