The ratio of the sums of m and n terms of an A.P. is m²: n². Show that the ratio of mth and nth term is (2m – 1): (2n – 1).
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Let S
Let S m
Let S m
Let S m and S
Let S m and S n
Let S m and S n
Let S m and S n be the sum of the first m and first n terms of the AP respectively. Let, a be the first term and d be a common difference
Let S m and S n be the sum of the first m and first n terms of the AP respectively. Let, a be the first term and d be a common differenceS
Let S m and S n be the sum of the first m and first n terms of the AP respectively. Let, a be the first term and d be a common differenceS n
Let S m and S n be the sum of the first m and first n terms of the AP respectively. Let, a be the first term and d be a common differenceS n 2
Let S m and S n be the sum of the first m and first n terms of the AP respectively. Let, a be the first term and d be a common differenceS n 2
Let S m and S n be the sum of the first m and first n terms of the AP respectively. Let, a be the first term and d be a common differenceS n 2 m
Let S m and S n be the sum of the first m and first n terms of the AP respectively. Let, a be the first term and d be a common differenceS n 2 m 2
Let S m and S n be the sum of the first m and first n terms of the AP respectively. Let, a be the first term and d be a common differenceS n 2 m 2
Let S m and S n be the sum of the first m and first n terms of the AP respectively. Let, a be the first term and d be a common differenceS n 2 m 2
2
2n
2n
2n [2a+(n−1)d]
2n [2a+(n−1)d]2
2n [2a+(n−1)d]2m
2n [2a+(n−1)d]2m
2n [2a+(n−1)d]2m [2a+(m−1)d]
2n [2a+(n−1)d]2m [2a+(m−1)d]
2n [2a+(n−1)d]2m [2a+(m−1)d]
2n [2a+(n−1)d]2m [2a+(m−1)d] n
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d2a+(m−1)d
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d2a+(m−1)d
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d2a+(m−1)d
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d2a+(m−1)d n
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d2a+(m−1)d nm
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d2a+(m−1)d nm
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d2a+(m−1)d nm
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d2a+(m−1)d nm n[2a+(m−1)d]=m[2a+(n−1)d]
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d2a+(m−1)d nm n[2a+(m−1)d]=m[2a+(n−1)d]2an+mnd−nd+2am+mnd−nd
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d2a+(m−1)d nm n[2a+(m−1)d]=m[2a+(n−1)d]2an+mnd−nd+2am+mnd−ndmd−nd=2am−2an
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d2a+(m−1)d nm n[2a+(m−1)d]=m[2a+(n−1)d]2an+mnd−nd+2am+mnd−ndmd−nd=2am−2an(m−n)d=2a(m−n)
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d2a+(m−1)d nm n[2a+(m−1)d]=m[2a+(n−1)d]2an+mnd−nd+2am+mnd−ndmd−nd=2am−2an(m−n)d=2a(m−n)d=2a
2n [2a+(n−1)d]2m [2a+(m−1)d] n 2 m 2 2a+(n−1)d2a+(m−1)d nm n[2a+(m−1)d]=m[2a+(n−1)d]2an+mnd−nd+2am+mnd−ndmd−nd=2am−2an(m−n)d=2a(m−n)d=2aNow, the ratio of mth and nth terms is
a+(n−1)d
a+(n−1)da+(m−1)d
a+(n−1)da+(m−1)d
a+(n−1)da+(m−1)d
a+(n−1)da+(m−1)d a+(n−1)2a
a+(n−1)da+(m−1)d a+(n−1)2aa+(m−1)2a
a+(n−1)da+(m−1)d a+(n−1)2aa+(m−1)2a
a+(n−1)da+(m−1)d a+(n−1)2aa+(m−1)2a
a+(n−1)da+(m−1)d a+(n−1)2aa+(m−1)2a
a+(n−1)da+(m−1)d a+(n−1)2aa+(m−1)2a a(1+2n−2)
a+(n−1)da+(m−1)d a+(n−1)2aa+(m−1)2a a(1+2n−2)a(1+2m−2)
a+(n−1)da+(m−1)d a+(n−1)2aa+(m−1)2a a(1+2n−2)a(1+2m−2)
a+(n−1)da+(m−1)d a+(n−1)2aa+(m−1)2a a(1+2n−2)a(1+2m−2)
a+(n−1)da+(m−1)d a+(n−1)2aa+(m−1)2a a(1+2n−2)a(1+2m−2)
a+(n−1)da+(m−1)d a+(n−1)2aa+(m−1)2a a(1+2n−2)a(1+2m−2) 2n−1
2m-1
Thus, ratio of its mth and nth terms is 2m−1:2n−1
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