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.

Let the initial price of the item be 100%.
We are given fifteen successive percentage discounts in the series: 2%, 4%, 6%, 8%, and so on. The n-th discount percentage is given by the formula:
$d\left(n\right)=2n\%$

After each discount, the remaining value of the item is multiplied by the factor:
$1-\frac{2n}{100}$

We want to find after how many discounts the effective discount is higher than 50%. This means the remaining percentage value of the item must fall below 50%.
Let us calculate the remaining percentage value step-by-step after each discount:

1. After the 1st discount (2%):
$100\times (1-0.02)=100\times 0.98=98\%$

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

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

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

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

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

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

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

Since the remaining value after the 8th discount is approximately 46.55%, the total effective discount is:
$100-46.55=53.45\%$
This is the first time the effective discount exceeds 50%.

Therefore, after the 8th percentage discount in succession, the effective discount will be higher than 50%.