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Let Tr be the rth term of an AP for r=1,2,3,.... If for some positive integers m, n. We have Tm=1/n and Tn=1/m, then Tmn equals to
![question image](http://images.topperlearning.com/topper/new-ate/top_mob1694856666233731742e209d7f4-743d-4f07-845e-7c78b5f76485.jpg)
Asked by nitinsolanki14102001 | 16 Sep, 2023, 15:01: PM
Let a be the initial value and d be the common difference of given arithmetic progression.
Let rth term of arithmetic progression Tr is
![begin mathsize 14px style T subscript r space equals space a space plus space left parenthesis r minus 1 right parenthesis space d end style](https://images.topperlearning.com/topper/tinymce/cache/910ffa352bed4c7c5f91d40cc44a808b.png)
Given mth and nth terms are
![begin mathsize 14px style T subscript m space equals space a space plus space left parenthesis m minus 1 right parenthesis space d space equals space 1 over m end style](https://images.topperlearning.com/topper/tinymce/cache/8b8c53c6d88f8c4ca0f2d89403d35f5b.png)
![begin mathsize 14px style T subscript n space equals space a space plus space left parenthesis n minus 1 right parenthesis space d space equals space 1 over n end style](https://images.topperlearning.com/topper/tinymce/cache/45515485535ba23d60ba32f59c66b21b.png)
By solving above simultaneous linear equations in a and d, we get
![begin mathsize 14px style a space equals space fraction numerator m plus n minus 1 over denominator m space n end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/9696ec8beaa1fc50da4ca71b33bc3b82.png)
![begin mathsize 14px style d space equals negative space fraction numerator 1 over denominator m space n end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/1effea630f51d0f734fdd800f4f207ef.png)
Then we get Tmn as
![begin mathsize 14px style T subscript m n end subscript space equals space a space plus space left parenthesis space m space n space minus space 1 space right parenthesis space d end style](https://images.topperlearning.com/topper/tinymce/cache/2ad90649e6a3a910f49a2f16023b1113.png)
![begin mathsize 14px style T subscript m n end subscript space equals space fraction numerator m plus n minus 1 over denominator m space n end fraction space minus space left parenthesis space m space n space minus 1 space right parenthesis space fraction numerator 1 over denominator m n end fraction space end style](https://images.topperlearning.com/topper/tinymce/cache/f066a2812554a6bbd073d8ce6fdf1293.png)
![begin mathsize 14px style T subscript m n end subscript space equals space fraction numerator m space plus space n space over denominator m n end fraction space minus space 1 end style](https://images.topperlearning.com/topper/tinymce/cache/3e2aae08e68b8c96e87bb9ae901854e2.png)
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