On the day Sara was born 500 was invested and it gained 6.4% interest=6.4% of 500=5*6.4=32 by the time she was one year old. After 25 years that will have multiplied by 25=25*32=800. When Sara was 1 another 500 was invested and after 24 years this would gain 24*32=768. When she was 2 there would be 23 years left and 23*32=736 would accumulate. So we have a progression 800+768+736+... This can be written 32(25+24+23+...1). The sum of the numbers between 1 and 25 is given by 25*26/2=13*25=325. (If you look at the series and take the first and last terms they add up to 26; then the sum of 24 and 2, 23 and 3, all add up to 26. In the middle we have 12+14 and 13 on its own, so we have 12 pairs of number adding up to 26=312 plus 13=325.) Multiply by 32: 32*325=10400. But her parents invested 500 a year for 25 years so we need to add 25*500 to this interest=10400+12500=22,900.
