The greatest integer less than or equal to the sum of first 100 terms of the sequence 1 3 , 5 9 , 19 27 , 65 81 , … is equal to ______

Question

The greatest integer less than or equal to the sum of first 100 terms of the sequence 1 3 , 5 9 , 19 27 , 65 81 , &#8230; is equal to ______

Aicharya · Accepted Answer

Let S = 1 3 + 5 9 + 19 27 + &#8943; upto 100 terms S = 3 - 2 3 1 + 3 2 - 2 2 3 2 + 3 3 - 2 3 3 3 + . . . . upto 100 terms So, T n = 1 &#8722; 2 3 n &#8658; S 100 = 100 &#8722; 2 3 1 &#8722; 2 3 100 1 &#8722; 2 3 = 98 + 2 2 3 100 &#8658; &#160;&#160; S 100 = 98

The greatest integer less than or equal to the sum of first 100 terms of the sequence 1 3 , 5 9 , 19 27 , 65…

Solution