Question

AP consists of three terms whose sum is 15 and sum of the square of extremes is 58 find the 3 terms of AP and also find the sum of first 50 terms of an ap​

Answer 1

I hope you will understand the answer .

Answer:

Sum of the first 50 terms is 2700.

Pls mark it as brainliest .

Answer 2

${\bigstar \; \mathfrak{\underline{Answer:}}}$  

The sum of first 50 terms of an A.P is 2700.

${\bigstar \; \mathfrak{\underline{Step - by - step \; explanation:}}}$  

Given as :  

For an A.P , the sum of three terms = 15  

Let The three A.P terms are,  a - d  ,  a  , a + d  

So, The sum of three terms = (a -d) + a + (a + d)  

Or,  (a -d) + a + (a + d) = 15  

Or, (a + a + a) + ( - d + d) = 15  

Or, 3 a + 0 = 15  

$\sf a = \dfrac{\cancel{ 15}}{\cancel{3}}$  

$\qquad \quad \therefore \underline{\boxed{\frak{\pink {a = 5}}}}$

So, The first term = a = 5  

Again,  

The sum of square of extremes = 58  

So, (a - d)² + (a + d)² = 58  

Or, a² - 2 a d + d² + a² + 2 a d + d² = 58  

Or, 2 a² + 2 d² + 0 = 58  

Or, a² + d² = $\sf \dfrac{58}{2}$  

Or, a² + d²  = 29  

Put the value of a  

So, 5² + d²  = 29  

Or, 25 + d²  = 29  

Or,  d²  = 29 - 25  

Or, d²  = 4  

$\sf \; d = \sqrt{4}$  

$\qquad \quad\therefore \underline{\boxed{\frak{\pink{d = 2}}}}$ 

So, The common difference in A.P = d = 2  

∴ The first term of A.P = a - d = 5 - 2

I.e. The first term of A.P = 3  

The second term of A.P = a  

i.e. The second term of A.P = 5  

The third term of A.P = a + d = 5 + 2  

i.e. The third term of A.P = 7  

Again

Let The sum of first 50 terms = s  

∵ The sum of n terms of A.P =   

So, The sum of first 50 terms = $\sf \dfrac{50}{2} \; \big[2 \times 5 + (50 - 1) 2\big]$  

Or, s = 25 × [10 + 49 × 2]  

Or, s = 25 × [108]  

$\qquad \quad \sf or \;\underline{\boxed{\frak{\pink{s = 2700}}}}$

So, The sum of first 50 terms of an A.P = s = 2700

 

Hence, The sum of first 50 terms of an A.P is 2700.

         ___________________