1. First of all, today is my 25th birthday and I begin thinking about retirement. My goal is to retire on my 67th birthday with enough money saved to meet my needs. The first payment is $350,000 and it should be increased by 2% annually during the thirty years of retirement. Thus, assuming the first payment is S, the second payment will be:

S2 = S * (1+0,02) = 350,000 * 1,02 = 357,000;

while the third payment is:

S3 = S2 * (1+0,02) = S*(1+0,02)*(1+0,02) = 350,000*(1+0,02)*(1+0,02)=364,140;

Similarly, the fourth payment will be:

S4 = S3*(1+0,02) = S*(1+0,02)*(1+0,02)*(1+0,02) = 371,422.8

Thus, the key formula is:

It can be considered as a geometric progression. Thus, its sum is calculated by using the following formula:

Thus, the total sum of payments during the thirty years is

S30 = 350,000*((-0,02)^30)/-0,02 = 14,198,827.72

Taking into account that I want to purchase a new car for $75,000 on my 75th birthday, the minimum amount of money I must have in my account on the 67th birthday is 14,198,827.72 + 75,000 = 14,273,827.72.

2. Secondly, I have to calculate the minimum annual deposit that should be made into my account, assuming that, the annual interest rate is 10%. The first sum should be made on my 26^{th} birthday, while the last one will be made on 67^{th} birthday. Thus, the total period of deposit is 42 years. Thus, the minimum amount of deposit can be calculated from the following formula:

14,273,827.72 = b*42*(1+0,1);

where b is a minimum amount of deposit. Thus, b = 14,273,827.72 / (1,01*42) = 308,957.31

That is why the minimum annual deposit that should be made is $308,957.31.