Question

ratio of 2nd term to 6th term is 2 :5 in an ap. 8th term is 26 what is the 10th term​

Answer 1

Answer:

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Answer 2

Given :-

Ratio of 2nd term to the 6th term is 2 : 5 .

8th term is 26 .

Required to find :-

• 10th term ?

Formula used :-

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

Solution :-

Given data :-

Ratio of 2nd term to 6th term is 2 : 5

8th term is 26

we need to find the 10th term

So,

From the given data let's find the values of a ,d

Here,

Ratio of 2nd term to 6th term is 2 : 5

2nd term : 6th term = 2 : 5

But,

2nd term can be represented as ;

a + d

Similarly,

6th term can be replaced as ;

a + 5d

This implies ;

a + d : a + 5d = 2 : 5

This can also be written as ;

$\tt{ \dfrac{a + d}{a + 5d} = \dfrac{2}{5}}$

By, cross multiplication

5 ( a + d ) = 2 ( a + 5d )

5a + 5d = 2a + 10d

5a - 2a + 5d - 10d = 0

3a - 5d = 0 $\longrightarrow{\tt{\bf{Equation-1}}}$

consider this as equation - 1

However,

It is also mentioned that ;

8th term = 26

So,

8th term is also represented as ; a + 7d

a + 7d = 26 $\longrightarrow{\tt{\bf{Equation-2}}}$

consider this as Equation 2

Now,

Multiply equation 2 with 3

3 ( a + 7d ) = 3 ( 26 )

3a + 21d = 78 $\longrightarrow{\tt{\bf{Equation-3}}}$

consider this as equation 3

Subtract equation 1 from Equation 3

$\tt{3a + 21d = 78} \\ \tt{3a - 5d = 0} \\ \underline{( - )( + ) \: \: \: ( - ) \: \: \: \: } \\ \underline{\: \: \: \: \: \: \tt + 26d = 78} \\ \\ \implies \tt 26d = 78 \\ \\ \implies \tt d = \dfrac{78}{26} \\ \\ \implies \tt d = 3$

substitute the value of d in equation 2

a + 7d = 26

a + 7 ( 3 ) = 26

a + 21 = 26

a = 26 - 21

a = 5

Hence,

• First term ( a ) = 5

• Common difference ( d ) = 3

Using the formula ;

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

$\longrightarrow{\tt{ {a}_{nth} = {a}_{10} }}$

$\longrightarrow{\tt{ {a}_{10} = 5 + ( 10 - 1 ) 3 }}$

$\longrightarrow{\tt{ {a}_{10} = 5 + ( 9 ) 3 }}$

$\longrightarrow{\tt{ {a}_{10} = 5 + 27 }}$

$\longrightarrow{\tt{ {a}_{10} = 32 }}$

Therefore,

10th term = 32