Prove 2 32 +22 33 +23 34 + +2n 3n+1 = 18/5 (6n -1) by the principle of mathematical - Maths - Principle of Mathematical Induction

Question

​Prove 2 32 +22 33
+23 34 +… +2n​
3​n+1 = 18/5 (6n -1) by the principle
of mathematical induction for every nєN

Neha Sethi · Accepted Answer

Dear student
Let the given statement be Pnie Pn:2
32+22 33+23 34+ +2n
3n+1=1856n-1For n=1, we haveP1:2n
3n+1=21×31+1=2×32=18and 18561-1=185×5=18 which is true
Let Pk be true for some postive integer k ie2
32+22 33+23 34+ +2k 3k+1=1856k-1     
(1)We shall now prove that Pk+1 is true2
32+22 33+23 34+ +2k 3k+1+2k+1 3k+1+1 =2 32+22 33+23 34+ +2k
3k+1+2k+1 3k+2=1856k-1+2k+1
3k+2     [using 1]=2×326k-1+5×2k+1
3k+25=2k+1×3k+2-2×32+5×2k+1
3k+25=2k+1×3k+21+5-185=2k+1×3k+2×6-185=2k+2×3k+3-185=2k×4×3k×27-185=6k×108-185=1856k×6-1=1856k+1-1Thus P(k+1) is true whenever  P(k) is true
Hence by PMI, statement P(n) is true for all n∈N
Regards

​Prove 2.32 +22.33 +23.34 +… +2n​.3​n+1 = 18/5 (6n -1) by the principle of mathematical induction for every nєN.

Prove 2.3² +2².3³ +2³.3⁴ +… +2ⁿ.3ⁿ⁺¹ = 18/5 (6ⁿ -1) by the principle of mathematical induction for every nєN.