Find the sum to n terms of the following series:52 + 62 + 72 
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    Sequences and Series

    Multiple Choice QuestionsShort Answer Type

    121. Find the sum of the squares of first n natural numbers. Or

    Evaluate  space space sum straight n squared space space space or space space space space sum from straight k equals 1 to straight n of space straight k squared
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    Multiple Choice QuestionsLong Answer Type

    122. Find the sum of the cubes of first n natural numbers.  Or

    Evaluate  space space sum straight n cubed space space or space space space sum from straight k space equals 1 to straight n of space straight k cubed  Also, show that sum straight n cubed space equals space left parenthesis sum straight n right parenthesis squared
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    Multiple Choice QuestionsShort Answer Type

    123. Find the sum of n terms of the series whose nth term is

    space space fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus...... plus straight n cubed over denominator straight n end fraction.
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    Multiple Choice QuestionsLong Answer Type

    124.

    Find the sum to n terms of the series:

    1 . 2 . 4 + 2. 3 . 7 + 3. 4. 10 +..............

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    Multiple Choice QuestionsShort Answer Type

    125. Find the sum of the cubes of first n odd natural numbers. Or Sum the series to n terms l3 + 33 + 53 +..........
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    126.

    Find the sum to n terms the series fraction numerator 1 over denominator 1.2 end fraction plus fraction numerator 1 over denominator 2.3 end fraction plus fraction numerator 1 over denominator 3.4 end fraction plus........... and deduce the sum to infinity.

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    127.

    Sum of the series n . 1 + (n - 1). 2 + (n - 2) . 3 + ........ + 1. n

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    Multiple Choice QuestionsLong Answer Type

    128. Show that the sum of the cubes of any number of consecutive integers is divisible by the sum of these integers.
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    Multiple Choice QuestionsShort Answer Type

    129.

    Show that fraction numerator 1 cross times 2 squared plus 2 cross times 3 squared plus...... plus straight n cross times left parenthesis straight n plus 1 right parenthesis squared over denominator 1 squared cross times 2 plus 2 squared cross times 3 plus...... plus straight n squared cross times left parenthesis straight n plus 1 right parenthesis end fraction space equals space fraction numerator 3 straight n plus 5 over denominator 3 straight n plus 1 end fraction

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    130.

    Find the sum to n terms of the following series:

    52 + 62 + 72 + ....+ 202. 

    space space space space space space space space space 5 squared plus 6 squared plus.... plus 20 squared space equals space left parenthesis 1 squared plus 2 squared plus 3 squared plus.... plus 20 squared right parenthesis space minus space left parenthesis 1 squared plus 2 squared plus 3 squared plus 4 squared right parenthesis

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                                        = space space space fraction numerator left parenthesis 20 right parenthesis space left parenthesis 21 right parenthesis space left parenthesis 41 right parenthesis over denominator 6 end fraction minus fraction numerator left parenthesis 4 right parenthesis space left parenthesis 5 right parenthesis space left parenthesis 9 right parenthesis over denominator 6 end fraction           open square brackets use space sum straight n squared space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction close square brackets

                                        = 2870 - 30 = 2840
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