In some country the average annual growth rate of GDP was 5% in each of the first 42 years, and in each of the next 35 years was 6%. By how much...

Hello!

If a constant annual growth rate `p%` of GDP is known, the rule of 70 gives an approximation of how many years n it will take to double GDP. Specifically,

`n approx 70/p.`

As you understand, actually `(1+p/100)^n=2,` take the natural logarithm of both sides and obtain  `n ln(1+p/100)=ln2,` or `n=(ln2)/ln(1+p/100).`  For small `p` 's it is approximately `(ln2)/((p/100))=(100 ln2)/p approx 70/p.`

In our problem, `p_1=5%,` so GDP will double after about `70/5=14` years. It will double `42/14=3` times, i.e. will be multiplied by `2*2*2=8.`

After that, `p_2=6%` and GDP will double after about `70/6=35/3` years. Thus GDP will double `35/((35/3))=3` times and will be also multiplied by `8.`
The combined growth will be `8*8 = 64` times (using the 70 rule). Actually it will be `(1+5/100)^42*(1+6/100)^35 approx 60` times.

Growth by 64 times is the same as growth by `(64-1)*100%` = 6300%. This is the answer.