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  • If the ratio of sum of first n terms of two A.P s is (7n+1):(4n+27), find the ratio of their <img alt="m ^{th }" src="https://entrancecorner.codecogs.com/gif.latex?m%20%5E%7Bth%20%

If the ratio of sum of first n terms of two A.P s is (7n+1):(4n+27), find the ratio of their m ^{th } term 

 

 

 

 
 
 
 
 

Answers (1)

\frac {S_n}{S'_n} = \frac {7n+1}{4n +27} \\\\

Let a,a' be the first terms and d , d' the common differences of the two given APs 

\frac {S_n}{S'_n} = \frac {7n+1}{4n +27} \\\\\\ \frac {n/2[2a+(n-1)d]}{n/2[ 2a'+ ( n-1 )d']} = \frac {7n+1}{4n +27} \\\\\\ \frac {2a+ ( n-1 )d}{2a' ( n-1)d'} = \frac {7n+1}{4n +27}

To find the ratio of m^{th} term replace n by (2m-1) 

\\\\\\=\frac {2a+ ( 2m-1 -1 )d}{2a' ( (2m-1)-1)d'} = \frac {7(2m-1)+1}{4(2m-1)+27}\\\\ = \frac{2a + ( 2m-2 )d}{2a' + ( 2m-1)d'}= \frac{14m-7+1}{8m -4 + 27 }\\\\

\frac { 2 [ a+ ( m-1)d]}{2[a' + ( m-1)d']} = \frac {14 m - 6}{8m - 4 + 27 } \\\\\\\\ \frac { a + ( m-1 )d}{a ' + ( m-)d'} = \frac { 14 m-6}{8m+23}\\\\\\\frac { a _m }{a _ m '} = \frac {14 m -6 }{8m + 23 }

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Ravindra Pindel

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