Question

1+3+5+......=n^2 prove by mathematical induction

Answer 1

Step-by-step explanation:

$P(n) = 1 + 3 + 5 + 7........ + (2n - 1) = {n}^{2} \\ When \:, n = 1 \\ L.H.S \: = \: \: 2 \times 1 - 1 = 1 \\ R.H.S \: \: = \: \: \: \: \: \: \: 1 \\ L.H.S = R.H.S \\ Hence \: P(1) \: is \: true. \\ Let \: P(k) \: be \: true \: \\ P(k) = 1 + 3 + 5..........(2k - 1) = {k}^{2} ...........(1) \\ To \: Prove \: P(k + 1) \: is \: true \: i.e. \\ 1 + 3 + 5 + ........ +( 2(k + 1) - 1) = {(k + 1)}^{2} \\ Adding \: (2k + 1) \: \: to \: both \: sides \: of \: (1) \\ 1 + 3 + 5 + ........ + (2k -1 ) + (2k + 1) = {k}^{2} + 2k + 1 \\ hence \: P(k + 1) \: is \: true \:, whenever \: P(k)is \: true. \\ \\ By \: principle \: of \: mathematical \: induction \: it \: follows \: that \: P(n) \\ is \: true \: for \: all \: natural \: numbers \: n. \\ \\ \\ Hope \: it \: will \: help \: you.$