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

The correct option is 8th.

To find the number of successive discounts needed to achieve an effective discount higher than 50%, we can calculate the remaining price of the item step-by-step. Let the original price of the item be 100.

Each discount in the successive series is defined by the progression: 2%, 4%, 6%, 8%, and so on. The n-th discount rate is given by $2n\%$.

When a percentage discount of d% is applied, the remaining price factor is multiplied by $(1-\frac{d}{100})$. For the effective discount to be higher than 50%, the remaining price must drop below 50% of the original price (i.e., less than 50).

Let's calculate the remaining price after each successive discount:

1. After the 1st discount (2%):
$$P_1=100\times (1-0.02)=98$$

2. After the 2nd discount (4%):
$$P_2=98\times (1-0.04)=98\times 0.96=94.08$$

3. After the 3rd discount (6%):
$$P_3=94.08\times (1-0.06)=94.08\times 0.94=88.435$$

4. After the 4th discount (8%):
$$P_4=88.435\times (1-0.08)=88.435\times 0.92\approx 81.36$$

5. After the 5th discount (10%):
$$P_5=81.36\times (1-0.10)=81.36\times 0.90\approx 73.22$$

6. After the 6th discount (12%):
$$P_6=73.22\times (1-0.12)=73.22\times 0.88\approx 64.44$$

7. After the 7th discount (14%):
$$P_7=64.44\times (1-0.14)=64.44\times 0.86\approx 55.42$$

8. After the 8th discount (16%):
$$P_8=55.42\times (1-0.16)=55.42\times 0.84\approx 46.55$$

At the 7th discount, the remaining price is approximately 55.42%, which means the cumulative discount is 44.58%.

At the 8th discount, the remaining price drops to approximately 46.55%, which corresponds to a cumulative discount of 53.45%.

Since 53.45% is higher than 50%, the effective discount becomes higher than 50% after the 8th successive percentage discount.