Hint :-
Solution :-
We need to find out the sum of 50 terms of the given sequence . After splitting the given sequence ,
[tex]\implies S_1 = \dfrac{1}{2},\dfrac{1}{4},\dfrac{1}{8}[/tex] .
We can see that this is in Geometric Progression where 1/2 is the common ratio . Calculating the sum of 25 terms , we have ,
[tex]\implies S_1 = a\dfrac{1-r^n}{1-r} \\\\\implies S_1 = \dfrac{1}{2}\left[ \dfrac{1-\bigg(\dfrac{1}{2}\bigg)^{25}}{1-\dfrac{1}{2}}\right] [/tex]
Notice the term [tex]\dfrac{1}{2^{25}}[/tex] will be too small , so we can neglect it and take its approximation as 0 .
[tex]\implies S_1\approx \cancel{ \dfrac{1}{2} } \left[ \dfrac{1-0}{\cancel{\dfrac{1}{2} }}\right] [/tex]
[tex]\\\implies \boxed{ S_1 \approx 1 }[/tex]
[tex]\rule{200}2[/tex]
Now the second sequence is in Arithmetic Progression , with common difference = 3 .
[tex]\implies S_2=\dfrac{n}{2}[2a + (n-1)d] [/tex]
Substitute ,
[tex]\implies S_2=\dfrac{25}{2}[2(4) + (25-1)3] =\boxed{ 908} [/tex]
Hence sum = 908 + 1 = 909
