Question

the sum of first 25 terms of the series 1+2+3+....​

Answer 1

Question :

Find the sum of first 25 terms of the Arithmetic series 1,2,3... !!

Given :

• No. of terms = 25
• Arithmetic series = 1,2,3,..

To Find :-

The sum of first 25 terms of the series.

Solution :-

According to the given information , we can determine the first term of the AP and it's common difference.

• First term = 1

• Common difference

We know that Common Difference is :

⠀⠀⠀⠀⠀⠀⠀$\underline{\bf{d = a_{n} - a_{n - 1}}}$

Now , using the formula and substituting the values in it, we get :

$:\implies \bf{d = a_{n} - a_{n - 1}}$

$:\implies \bf{d = 3 - 2}$

$:\implies \bf{d = 1}$

$\therefore \bf{Common\:Difference = 1}$

Hence, common difference of the AP is 1 .

Now , we have to find the last term of the AP :

By using the formula for nth term and Substituting the values in it, we get :

$:\implies \underline{\bf{t_{n} = a_{1} + (n - 1)d}}$

$:\implies \bf{t_{n} = 1 + (25 - 1) \times 1}$

$:\implies \bf{t_{n} = 1 + 24 \times 1}$

$:\implies \bf{t_{n} = 1 + 24}$

$:\implies \bf{t_{n} = 25}$

Hence, the 25th term of the AP is 25.

⠀⠀⠀⠀To find the sum of 25 terms , we get :

We know the formula for sum of n terms i.e,

$\underline{\boxed{\bf{s_{n} = \dfrac{n}{2}\bigg(a + l\bigg)}}}$

$:\implies \bf{s_{n} = \dfrac{25}{2}\bigg(1 + 25\bigg)}$

$:\implies \bf{s_{n} = \dfrac{25}{2} \times 26}$

$:\implies \bf{s_{n} = 25 \times 13}$

$:\implies \bf{s_{n} = 325}$

Hence the sum of 25 terms of the AP is 325.

Answer 2

Answer:

Sum of 1st 25 th terms is 325.

Step-by-step explanation:

Given series is 1+2+3+4+5+....+25

It is clear that it is a AP series because here difference between all consecutive terms is 1.

We know Sum in AP series = $S = \frac{n}{2} [2a + (n − 1) × d].....(1)$

Here $a$is the 1S term of the series.

d is common difference and n is number of terms.

From equation (1)

Required Sum=$S = 25/2[2 \times 1 + (25 − 1) × 1] = \frac{25}{2} (2 + 24) = 25 \times 13 = 325$

So, Required sum is 325.