expand (1+5x)^7 in ascending powers of x up to and including the term x^2.?

Question

By putting x=0.01, find an approximation for 1.05^7.

*please do not say "thats chating do your own work". I obviously would do so if i could, however i just don't know how to do this :(

Answer 1

(1+5x)^7
=1+7(5x)+(7x6/2)(25x^2)+...
=1+ 35x + 525x^2 +...

1.05^7
=(1+5(0.01))^7
= 1+ 35(0.01) + 525(0.01)^2 + ...
= 1 + 0.35 + 0.0525 + ...
= 1.4025 (approximate)

Answer 2

To find an approximation of 1.05^7, I'd just use a calculator.
The process of multiplying (1+5x) times itself seven times means multiplying:
(1+5x) (1+5x) [ = 1 + 10x +25]
Multiply 1+10x+25x^2 by (1+5) [get that product and repeat]
You have to just multiply out the requisite number of times. Each time you will have a new product. Finally, you could check your answer (after you evaluate it) against what the calculator gives you.
After all the products are multiplied out, the expanded expression is:
1 + 35x +525x^2 +4375x^3 +21875x^4 + 65625x^5 + 109375x^6 +78125x^7