Necessary Condition for an Extrema(Is a Function Differentiable at Boundaries)
Let ∑ k =110 ...
Question
Let 10∑k=1f(a+k)=16(210−1), where the function f satisfies f(x+y)=f(x)+f(y) for all natural numbers x,y and f(1)=2. Then the natural number 'a' is:
A
2
No worries! We‘ve got your back. Try BYJU‘S free classes today!
B
3
Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses
C
4
No worries! We‘ve got your back. Try BYJU‘S free classes today!
D
16
No worries! We‘ve got your back. Try BYJU‘S free classes today!
Open in App
Solution
The correct option is B3 We have f(1)=2andf(x+y)=f(x)+f(y). ⇒f(1+1)=f(1)⋅f(1)=2⋅2=22f(1+2)=f(1)⋅f(2)=2⋅4=23⋮f(1+9)=210⇒f(x)=2x∀x∈N⇒2a+1+2a+2+⋯+2a+10=16(210−1)⇒2a(2+22+⋯+210)=16(210−1)⇒2a⋅2⋅(210−1)1=16(210−1) On comparing, we get a+1=4 Hence, a=3.