Question

find the sum of first 25 terms of ap -2, -5, -8, -11​

Answer 1

$\large\bf\underline{Given:-}$

• AP = -2, -5 , -8 ,-11

$\large\bf\underline {To \: find:-}$

• sum of 25 terms

$\huge\bf\underline{Solution:-}$

≫ Given AP = -2 , -5 ,-8 , -11

• ➣ First term = -2
• ➣ common difference = -5-(-2) = -3
• ➣ Number of terms = 25

$\bullet \: \large \bf \: S_n = \frac{n}{2} [2a + (n - 1)d]$

$\rm \rightarrowtail \:s_{25} = \frac{25}{2} \{2 \times( - 2) + 24 \times (- 3 ) \} \\ \\ \rm \rightarrowtail \:s_{25} = \frac{25}{2} \{ - 4 - 72 \} \\ \\ \rm \rightarrowtail \:s_{25} = \frac{25}{2} \{ - 76\} \\ \\ \rm \rightarrowtail \:s_{25} = \frac{25}{ \cancel2} \times ( \cancel{ - 76}) \\ \\ \rm \rightarrowtail \:s_{25} = 25 \times ( - 38) \\ \\ \rm \rightarrowtail \:s_{25} = - 950$

So,

≫Sum of first 25 terms of given AP is -950

Answer 2

Given :-

Arithmetic progession = - 2 , - 5 , - 8 , - 11 . . . . . . .

Required to find :-

• Sum of the first 25 terms

Formula used :-

$\large{\dagger{\boxed{\rm{{a}_{nth} = a + ( n - 1 ) d }}}}$

$\large{\dagger{\boxed{\rm{ {S}_{nth} = \dfrac{n}{2}[first \; term + last \ term ] }}}}$

Solution :-

Consider the given arithmetic progession ,

AP = - 2 , - 5 , - 8 , - 11 . . . . . . .

So,

• >> First term ( a ) = - 2

Common difference = ( 2nd term - 1st term ) = ( 3rd term - 3nd term )

=> ( - 5 - ( - 2 ) = ( - 8 - ( - 5 )

=> ( - 5 + 2 ) = ( - 8 + 5 )

=> ( - 3 ) = ( - 3 )

Hence,

• >> Common difference ( d ) = - 3

Now,

Using the formula ,

$\large{\dagger{\boxed{\rm{{a}_{nth} = a + ( n - 1 ) d }}}}$

( This formula enables us to find the 25th term )

>> Substitute the required values <<

$\longrightarrow{\sf{ {a}_{nth} = {a}_{25} }}$

$\longrightarrow{\sf{ {a}_{25} = - 2 + ( 25 - 1 ) - 3 }}$

$\longrightarrow{\sf{ {a}_{25} = - 2 + ( 24 ) - 3 }}$

$\longrightarrow{\sf{ {a}_{25} = -2 + ( - 72 ) }}$

$\longrightarrow{\sf{ {a}_{25} = - 2 - 72 }}$

$\longrightarrow{\sf{ {a}_{25} = - 74 }}$

• 25th term = - 74

The last term till which we want to find the sum is - 74 .

Hence,

Using the formula,

$\large{\dagger{\boxed{\rm{ {S}_{nth} = \dfrac{n}{2}[first \; term + last \ term ] }}}}$

Here,

$\Rightarrow{\tt{ {S}_{nth} = {S}_{25} }}$

$\Rightarrow{\tt{ {S}_{25} = \dfrac{25}{2} [ -2 + ( - 74 ) ]}}$

$\Rightarrow{\tt{ {S}_{25} = \dfrac{25}{2} [ - 2 - 74 ] }}$

$\Rightarrow{\tt{ {S}_{25} = \dfrac{ 25}{2} [ - 76 ] }}$

$\Rightarrow{\tt{ {S}_{25} = \dfrac{25}{\cancel{2}} \times {\cancel{ - 76 }}^{\large{- 38 }} }}$

$\Rightarrow{\tt{ {S}_{25} = 25 \times - 38 }}$

$\Rightarrow{\tt{ {S}_{25} = - 950 }}$

$\text{\underline{\underline{ Sum of first 25 terms = - 950 }}}{\checkmark}$