Different A.P’s are constructed with the first term 100, the last term 199 and integral common differences. The sum of the common differences of all such A.P.’s having at least 3 terms and at most 33 terms is……..

<p><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mn>9</mn><mn>9</mn><mo>−</mo><mn>1</mn><mn>0</mn><mn>0</mn></mrow><mrow><mn>2</mn></mrow></mfrac><menclose notation="updiagonalstrike"><mo>∈</mo></menclose><mi>l</mi></mrow></math></p><p><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mn>9</mn><mn>9</mn><mo>−</mo><mn>1</mn><mn>0</mn><mn>0</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>=</mo><mn>3</mn><mn>3</mn></mrow></math></p><p><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mn>9</mn><mn>9</mn><mo>−</mo><mn>1</mn><mn>0</mn><mn>0</mn></mrow><mrow><mn>4</mn></mrow></mfrac><menclose notation="updiagonalstrike"><mo>∈</mo></menclose><mi>l</mi></mrow></math></p><p><math><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn><mn>9</mn><mn>9</mn><mo>−</mo><mn>1</mn><mn>0</mn><mn>0</mn></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>∈</mo><mi>l</mi></mrow></math></p><p><math><mrow><mo>⇒</mo><mi>d</mi><mi>i</mi><mo>=</mo><mn>3</mn><mn>3</mn><mo>+</mo><mn>1</mn><mn>1</mn><mtext> </mtext><mtext> </mtext><mi>o</mi><mi>r</mi><mtext> </mtext><mtext> </mtext><mn>9</mn></mrow></math></p><p><math><mrow><mo>∴</mo></mrow></math>&nbsp;sum of common differences = 33 + 11 + 9 = 53</p>

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