Question

Find the sum of first 22 term of an AP whose 8 and 16 term are respectively 37 and 85​

Answer 1

Answer :-

Given :-

8th and 16th terms of the A.P are 37 & 85

Required to find :-

• Sum of first 22 terms of the A.P

Formula used :-

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

Solution :-

Given that :-

The 8th and 16th term of the A.P are 37 & 85

We need to find the sum of first 22 terms of the A.P

So,

8th term = 37

16th term = 85

But we know that

8th term is represented as a + 7d

So,

a + 7d = 37 $\longrightarrow{\text{Equation 1 }}$

Similarly ,

16th term is represented as a + 15d

So,

a + 15d = 85 $\longrightarrow{\text{Equation 2 }}$

Subtract equation 1 from equation 2

So,

a + 15d = 85

a + 7d = 37

(-) (-) (-)

$\rule{75}{1}$

0 + 8d = 48

$\rule{75}{1}$

This implies,

=> 8d = 48

=> d = 48/8

=> d = 6

Now, substitute the value of d in equation 1

So,

a + 7d = 37

a + 7 ( 6 ) = 37

a + 42 = 37

a = 37 - 42

a = - 5

Hence,

• value of " a " is - 5

• value of " d " is 6

Using the formula ,

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

Here,

a = first term

d = common difference

n = the term number which you want to find

$\rightarrowtail{\tt{ {S}_{nth} = {S}_{22} }}$

$\rightarrowtail{\tt{ {S}_{22} = \dfrac{22}{2}[ 2(-5) + ( 22 - 1 ) 6 ]}}$

$\rightarrowtail{\tt{ {S}_{22} = \dfrac{22}{2} [ - 10 + ( 21 ) 6 ]}}$

$\rightarrowtail{\tt{ {S}_{22} = \dfrac{22}{2} [ - 10 + 126 ]}}$.

$\rightarrowtail{\tt{ {S}_{22} = \dfrac{22}{2} [ 116 ] }}$

$\rightarrowtail{\tt{ {S}_{22} = 11 \times 116 }}$

$\rightarrowtail{\tt{ {S}_{22} = 1, 276 }}$

Therefore ,

Sum of first 22 terms = 1, 276

Answer 2

Given:–

a₈ = 37 and a₁₆ = 85

To find:—

The sum of first 22 terms

Solution:—

a₈ = a + 7d= 37⠀⠀⠀⠀⠀⠀⠀...①

a₁₆ = a + 15d= 85⠀⠀⠀⠀⠀⠀...②

Subtracting ① from ②, we get:-

8d = 48

d = 48/8 = 6

Common difference (d) = 6

Putting d = 6 in①:-

a + 42 = 37

a = 37 - 42 = -5

First term (a) = -5

Sn = n/2{2a + (n-1)d}

Here n= 22

S₂₂ = 22/2 {2 × (-5) + (22-1)6}

S₂₂ = 11 (-10 + 126)

S₂₂ = 11 × 116

S₂₂ = 1276

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