Question

নীচের শ্রেণির n সংখ্যক পদ পর্যন্ত সমষ্টি নির্ণয় করাে:
1 • 3 • 5 + 3 •5 • 7 + 5 • 7•9 + ....​

Answer 1

Answer :

S(n) = n(n + 2)(2n² + 4n - 1)

Solution :

Here ,

The given series is ;

1•3•5 + 3•5•7 + 5•7•9 + . . . up to n terms .

Clearly ,

The 1st factors , the 2nd factors and the 3rd factors of all the terms of the given series are in AP .

• Let's find 1st factor of nth term .

a = 1 , d = 3 - 1 = 2

Now ,

=> a(n) = a + (n - 1)d

=> a(n) = 1 + (n - 1)2

=> a(n) = 1 + 2n - 2

=> a(n) = 2n - 1

Hence ,

1st factor of nth term is (2n - 1) .

• Let's find 2nd factor of nth term .

a = 3 , d = 5 - 3 = 2

Now ,

=> a(n) = 3 + (n - 1)d

=> a(n) = 3 + (n - 1)2

=> a(n) = 3 + 2n - 2

=> a(n) = 2n + 1

Hence ,

2nd factor of nth term is (2n + 1) .

• Let's find 3rd factor of nth term .

a = 5 , d = 7 - 5 = 2

Now ,

=> a(n) = 5 + (n - 1)d

=> a(n) = 5 + (n - 1)2

=> a(n) = 5 + 2n - 2

=> a(n) = 2n + 3

Hence ,

2nd factor of nth term is (2n + 3) .

Now ,

The nth term of the given series will be given as ;

=> a(n) = 1st factor • 2nd factor • 3rd factor

=> a(n) = (2n - 1)(2n + 1)(2n + 3)

=> a(n) = [(2n)² - 1²](2n + 3)

=> a(n) = (4n² - 1)(2n + 3)

=> a(n) = 8n³ + 12n² - 2n - 3

Hence ,

The nth term of the given series is ;

a(n) = 8n³ + 12n² - 2n - 3

(Please refer to the attachment for further calculations)