There are fifteen successive percentage discounts given in a series of 2%, 4%, 6%, 8% and so on an item. After how many such percentage discounts in succession will the effective discount be higher than 50%?

The effective discount crosses 50% after the 8th discount.

Each discount leaves a factor (1 - 2k/100) for k = 1, 2, 3, ...

Multiply the factors step by step.

After 7 discounts the product is about 0.554, an effective discount of 44.6%.

After 8 discounts it falls to about 0.465, an effective discount of 53.5%.

Since 53.5% > 50%, the 8th discount is the first to exceed 50%.