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Sequences and Series

Short 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

129 Views

Long 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

110 Views

Short 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.

128 Views

Long Answer Type

124.

Find the sum to n terms of the series:

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

161 Views

Short 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 +..........

90 Views

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.

102 Views

127.

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

94 Views

Long Answer Type

128. Show that the sum of the cubes of any number of consecutive integers is divisible by the sum of these integers.

154 Views

Short Answer Type

129.
Show that $fraction numerator 1 cross times 2 squared plus 2 cross times 3 squared plus...... plus straight n cross times left parenthesis straight n plus 1 right parenthesis squared over denominator 1 squared cross times 2 plus 2 squared cross times 3 plus...... plus straight n squared cross times left parenthesis straight n plus 1 right parenthesis end fraction space equals space fraction numerator 3 straight n plus 5 over denominator 3 straight n plus 1 end fraction$

L.H.S = $fraction numerator 1 cross times 2 squared plus 2 cross times 3 squared plus......... plus straight n cross times left parenthesis straight n plus 1 right parenthesis squared over denominator 1 squared cross times 2 plus 2 squared cross times 3 plus..... plus straight n squared cross times left parenthesis straight n plus 1 right parenthesis end fraction space equals space fraction numerator sum straight n left parenthesis straight n plus 1 right parenthesis squared over denominator sum straight n squared cross times left parenthesis straight n plus 1 right parenthesis end fraction space equals space fraction numerator sum straight n left parenthesis straight n squared plus 2 straight n plus 1 right parenthesis over denominator sum straight n squared left parenthesis straight n plus 1 right parenthesis end fraction$

= $fraction numerator sum left parenthesis straight n cubed plus 2 straight n squared plus straight n right parenthesis over denominator sum left parenthesis straight n cubed plus straight n squared right parenthesis end fraction space equals space fraction numerator sum straight n cubed plus 2 sum straight n squared plus sum straight n over denominator sum straight n cubed plus sum straight n squared end fraction space equals space fraction numerator begin display style fraction numerator straight n squared left parenthesis straight n plus 1 right parenthesis squared over denominator 4 end fraction end style plus begin display style fraction numerator 2 straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction end style plus begin display style fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction end style over denominator begin display style fraction numerator straight n squared left parenthesis straight n plus 1 right parenthesis squared over denominator 4 end fraction end style plus begin display style fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction end style end fraction$

= $fraction numerator begin display style fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction end style open square brackets begin display style fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction end style plus begin display style fraction numerator 2 left parenthesis 2 straight n plus 1 right parenthesis over denominator 3 end fraction end style plus begin display style 1 over 1 end style close square brackets over denominator begin display style fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction end style open square brackets begin display style fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction end style plus begin display style fraction numerator 2 straight n plus 1 over denominator 3 end fraction end style close square brackets end fraction space equals space fraction numerator 3 straight n squared plus 3 straight n plus 8 straight n plus 4 plus 6 over denominator 3 straight n squared plus 3 straight n plus 4 straight n plus 2 end fraction$

= $space space space space space fraction numerator 3 straight n squared plus 11 straight n plus 10 over denominator 3 straight n squared plus 7 straight n plus 2 end fraction space equals space fraction numerator 3 straight n squared plus 6 straight n plus 5 straight n plus 10 over denominator 3 straight n squared plus 6 straight n plus straight n plus 2 end fraction equals space fraction numerator 3 straight n left parenthesis straight n plus 2 right parenthesis space plus space 5 left parenthesis straight n plus 2 right parenthesis over denominator 3 straight n left parenthesis straight n plus 2 right parenthesis plus 1 left parenthesis straight n plus 2 right parenthesis end fraction$

= $fraction numerator left parenthesis straight n plus 2 right parenthesis space left parenthesis 3 straight n plus 5 right parenthesis over denominator left parenthesis straight n plus 2 right parenthesis space left parenthesis 3 straight n plus 1 right parenthesis end fraction space equals space fraction numerator 3 straight n plus 5 over denominator 3 straight n plus 1 end fraction$