MTH202 Final Term Practice Questions

MTH202 Practice Questions 23-45

BS Computer science (Virtual University of Pakistan)

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MTH202 Practice Questions 23-45

BS Computer science (Virtual University of Pakistan)

Scan to open on Studocu

Studocu is not sponsored or endorsed by any college or university

Solution File Lecture No 23-25

Q1. Use mathematical induction to prove that 4n 3n 4 is true for integral values of n 2

Solution: · Let P(n) = 4n 3n 4 · Let P n 4n 3n 4 · n 2 · So P 2 · is true · 4k 3k 4 · Now · 4k 1 3k 1 4 · 16 4k 1 3k.(3 1) 16 · 4k 1 3k.3 3k 16 · 4k 1 3k 1 3k 4 12 · 4k 1 3k 1 4 12 3k

by (1) for k 2.

4k 1 3k 1 4 · ignoring 12 · 3k · as if a b c then a b

Hence it is proved by mathematical induction that 4n 3n 4 is true for integral values of n 2 .

Q3. Suppose n3 n is divisible by 6 is true for any positive integer. Justify the statement for n = k+1.

Solution: · Suppose for n=k, k 3 k is dividible by · 6 then k 3 k 6.q · we need to prove k 1 · 3 k 1 is dividible by 6 · k 1 · 3 k 1 k 3 3k 2 3k 1 k 1 · k 3 k 3k k 1 · 6.q 3k k 1 · As the · product · k k 1 of integers is even so we can take as 2r · 6.q 3k k 1 · 6.q 3.2r · 6 q r · So · k 1 · 3 k 1 is dividible by 6 · Hence by induction method, n3 n is divisible 6