Question 2:
Express 2n + 2 – 2n + 1 + 2n – 1 in the form p(2n – 1), where p is a constant.
Hence, solve the equation 8(2n + 2 – 2n + 1 + 2n – 1) = 5(2n2).
[6 marks]
Solution:
2n+2−2n+1+2n−1=(2n×22)−(2n×21)+(2n÷21)=2n(4−2+12)=2n(52)=5(2n2)=5(2n−1)∴p=5
8(2n+2−2n+1+2n−1)=5(2n2)8(5)(2n−1)=5(2)n223×2n−1=2n23+n−1=n2n2−n−2=0(n+1)(n−2)=0n=−1 or n=2
Express 2n + 2 – 2n + 1 + 2n – 1 in the form p(2n – 1), where p is a constant.
Hence, solve the equation 8(2n + 2 – 2n + 1 + 2n – 1) = 5(2n2).
[6 marks]
Solution:
2n+2−2n+1+2n−1=(2n×22)−(2n×21)+(2n÷21)=2n(4−2+12)=2n(52)=5(2n2)=5(2n−1)∴p=5
8(2n+2−2n+1+2n−1)=5(2n2)8(5)(2n−1)=5(2)n223×2n−1=2n23+n−1=n2n2−n−2=0(n+1)(n−2)=0n=−1 or n=2