Question

Let f(n) = `[1/3 + (3n)/100]n`, where [n] denotes the greatest integer less than or equal to n. Then `sum_(n = 1)^56f(n)` is equal to ______.

Options

• 56

• 689

• 1287

• 1399

Answer 1

Let f(n) = `[1/3 + (3n)/100]n`, where [n] denotes the greatest integer less than or equal to n. Then `sum_(n = 1)^56f(n)` is equal to 1399.

Explanation:

f(n) = `[1/3 + (3n)/100]n` 

f(1) = `[1/3 + (3 xx 1)/100](1)`

= [0.363] × 1

= 0 × 1

= 0

f(2) = [0.393] × 2

=  0 × 2

= 0

f(22) = `[1/3 + (3 xx 22)/100] xx 22`

= [0.993] × 22

f(1) + f(2) ........ + f(22) = 0

f(23) + f(24) + f(25) + ........ f(56) = 0

f(23) = `[1/3 + (3 xx 23)/100]23` = 23

f(24) = 24

f(55) = 55

f(56) = `[1/3 + (3 xx 56)/100] xx 56`

= `[(100 + 9 xx 56)/300] xx 56`

= 2 × 56

= 0 + 23 + 24 + 25 + ........ 55 + 2 × 56

= 1399