Question

find the 31st term of an ap whose 11th term is 38 and the16th term is 73.​

Answer 1

★ AnswEr:-

• 31st term is 178

★ GivEn:-

• The 11th term of A.P. is 38
• The 16th term of A.P is 73

★ To Find :-

• The 31st term of A.P.

$\sf \huge \red{Solution:-}$

We know, the formula of nth term of A.P,

$\sf \: T_{n} = a + (n - 1)d$

• a = 1st term
• d = common difference

According to the 1st condition,

$\sf \: T_{11} = a + (11 - 1)d = 38$

$\implies \: \sf a + 10d = 38......(i)$

According to the 2nd condition,

$\sf \: T_{16} = a + (16 - 1)d = 73$

$\implies \: \sf a + 15d = 73......(ii)$

From equation (i) and (ii) we get,

$\sf a + 15d = 73 \\ \sf a + 10d = 38 \\ ( - ) \: \: \: \: ( - ) \: \: \: \: ( - ) \\ ....................... \\ \sf \: \: \: \: \: \: \: \: \: 5d \: \: = 35$

$\boxed{ \red{ \sf{ \rightarrow \: d = \frac{35}{5} = 7}}}$

Now, putting the value of d in equation (i) we get,

$\implies \sf \: a + 10 \times 7 = 38$

$\implies \sf \: a + 70 = 38$

$\implies \sf \: a = 38 - 70$

$\boxed{ \red{ \sf{ \rightarrow \: a = - 32}}}$

Now, putting the values, in the formula we get,

$\sf \: T_{31} = a + (31 - 1)d$

$\implies\sf \: T_{31} = ( - 32) + 30 \times 7$

$\implies\sf \: T_{31} = ( - 32) + 210$

$\boxed{ \red{\implies{\sf \: T_{31} = 178}}}$

Answer 2

Step-by-step explanation:

Given,

•           11th term of the AP is 38

•           16th term of the AP is 73

To Find :

• find the 31st term of an ap

Solution :

From the properties of AP :

nth term = a + ( n - 1 )d      

 { where a is the first term and d is the common difference between the terms }

Let the first term of this AP be a and common difference between the terms be d.

⇒ 11th term = a + ( 11 - 1 )d

⇒ 38           = a + 10d             ...( 1 )

⇒ 16th term = a + ( 16 - 1 )d

⇒ 73            = a + 15d            ...( 2 )

Subtracting ( 2 ) from ( 1 ) :

⇒ a + 10d - ( a + 15d ) = 38 - 73

⇒ a + 10d - a - 15d = - 35

⇒ - 5d = - 35

⇒ d = 7

Hence,

⇒ 38 = a + 10d

⇒ 38 = a + 10( 7 )

⇒ 38 = a + 70

⇒ 38 - 70 = a

⇒ - 32 = a

Therefore,

⇒ 31st term = a + ( 31 - 1 )d

                   = - 32 + 30( 7 )

                   = - 32 + 210

                   = 178

Hence 31st term of the AP is 178.