Maths-
General
Easy

Question

1.2.3+2.3.4+3.4.5+........n terms. Find sum to n terms.

  1. fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis n plus 2 right parenthesis left parenthesis 3 n plus 5 right parenthesis over denominator 2 end fraction    
  2. fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis n plus 2 right parenthesis left parenthesis n plus 3 right parenthesis over denominator 4 end fraction    
  3. 2n(n + 1)(n + 2)
  4. fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis n plus 2 right parenthesis left parenthesis 3 n plus 1 right parenthesis over denominator 12 end fraction    

hintHint:

General Term (T subscript r space right parenthesis equals spacer left parenthesis r space plus 1 right parenthesis left parenthesis r plus 2 right parenthesis space equals r left parenthesis r squared space plus space 3 r space plus 2 right parenthesis space equals r to the power of 3 space end exponent plus space 3 r squared space plus space 2 r

The correct answer is: fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis n plus 2 right parenthesis left parenthesis n plus 3 right parenthesis over denominator 4 end fraction


    T subscript n space end subscript equals space open square brackets n to the power of t h end exponent space t e r m space o f space 1.2.3 close square brackets open square brackets n to the power of t h end exponent space t e r m space o f space 2.3.4 close square brackets open square brackets n to the power of t h end exponent space t e r m space o f space 3.4.5 close square brackets
W e space k n o w space t h a t space a subscript n space end subscript equals space a space plus space left parenthesis n minus 1 right parenthesis d
T subscript n space space end subscript equals space open square brackets 1 space plus open parentheses n minus 1 close parentheses 1 close square brackets open square brackets 2 space plus open parentheses n minus 1 close parentheses 1 space close square brackets open square brackets 3 space plus open parentheses n minus 1 close parentheses 1 space close square brackets
T subscript n space space end subscript equals space open square brackets 1 space plus open parentheses n minus 1 close parentheses close square brackets open square brackets 2 space plus open parentheses n minus 1 close parentheses space close square brackets open square brackets 3 space plus open parentheses n minus 1 close parentheses space close square brackets
T subscript n space space end subscript equals space left square bracket left parenthesis n right parenthesis open parentheses n plus 1 close parentheses left parenthesis n plus 2 right parenthesis right square bracket
W e space h a v e space t o space f i n d space o u t space S subscript n
S subscript n space equals begin inline style stack sum T subscript n with blank below end style space equals space begin inline style stack sum left square bracket space left parenthesis n right parenthesis left parenthesis n plus 1 right parenthesis left parenthesis n plus 2 right parenthesis right square bracket space equals space sum for blank of left square bracket left parenthesis n right parenthesis space left parenthesis n squared space plus 3 n space plus space 2 right parenthesis right square bracket with blank below end style
begin inline style stack S subscript n space equals space sum with blank below end style T subscript n begin inline style space end style begin inline style equals end style begin inline style space end style begin inline style sum for blank of end style begin inline style n cubed end style begin inline style space end style begin inline style plus end style begin inline style space end style begin inline style sum for blank of end style begin inline style 3 end style begin inline style n squared end style begin inline style space end style begin inline style plus end style begin inline style space end style begin inline style sum for blank of end style begin inline style 2 end style begin inline style n end style begin inline style blank end style begin inline style stack S subscript n space equals sum T subscript n with blank below end style begin inline style space end style begin inline style equals end style begin inline style space end style begin inline style sum for blank of end style begin inline style n cubed end style begin inline style space end style begin inline style plus end style begin inline style space end style begin inline style stack 3 sum with blank below end style begin inline style n squared end style begin inline style space end style begin inline style plus end style begin inline style space end style begin inline style stack 2 sum with blank below end style begin inline style n end style begin inline style blank end style
begin inline style W end style begin inline style e end style begin inline style space end style begin inline style k end style begin inline style n end style begin inline style o end style begin inline style w end style begin inline style space end style begin inline style t end style begin inline style h end style begin inline style a end style begin inline style t end style begin inline style space end style begin inline style S subscript n end style begin inline style space end style begin inline style equals end style begin inline style space end style n over 2 begin inline style left parenthesis end style begin inline style 2 end style begin inline style a end style begin inline style space end style begin inline style plus end style begin inline style space end style begin inline style left parenthesis end style begin inline style n end style begin inline style minus end style begin inline style 1 end style begin inline style right parenthesis end style begin inline style d end style begin inline style right parenthesis end style begin inline style blank end style
begin inline style S subscript n end style begin inline style space end style begin inline style equals end style begin inline style space end style begin inline style open square brackets begin display style fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style close square brackets squared end style begin inline style space end style begin inline style plus end style begin inline style space end style begin inline style 3 end style begin inline style open square brackets begin display style fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis over denominator 6 end fraction end style close square brackets end style begin inline style space end style begin inline style plus end style begin inline style 2 end style begin inline style open square brackets begin display style fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style close square brackets end style begin inline style blank end style
begin inline style S subscript n end style begin inline style space end style begin inline style equals end style begin inline style space end style begin inline style open square brackets begin display style fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style close square brackets squared end style begin inline style space end style begin inline style plus end style begin inline style space end style begin inline style open square brackets begin display style fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis over denominator 2 end fraction end style close square brackets end style begin inline style space end style begin inline style plus end style begin inline style open square brackets n left parenthesis n plus 1 right parenthesis close square brackets end style begin inline style blank end style
begin inline style T end style begin inline style a end style begin inline style k end style begin inline style i end style begin inline style n end style begin inline style g end style begin inline style space end style begin inline style o end style begin inline style u end style begin inline style t end style begin inline style space end style begin inline style c end style begin inline style o end style begin inline style m end style begin inline style m end style begin inline style o end style begin inline style n end style begin inline style blank end style
begin inline style S subscript n end style begin inline style space end style begin inline style equals end style begin inline style space end style begin inline style fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style begin inline style space end style begin inline style open square brackets begin display style fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style space plus space left parenthesis 2 n plus 1 right parenthesis space plus space 2 close square brackets end style begin inline style blank end style
begin inline style S subscript n end style begin inline style space end style begin inline style equals end style begin inline style space end style begin inline style fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction end style begin inline style space end style begin inline style open square brackets begin display style fraction numerator n squared space plus space n space space plus 2 left parenthesis 2 n space plus space 1 right parenthesis space plus space 2.2 over denominator 2 end fraction end style close square brackets end style begin inline style blank end style
begin inline style S subscript n end style begin inline style space end style begin inline style equals end style begin inline style space end style fraction numerator n left parenthesis n plus 1 right parenthesis open square brackets n squared space plus space n space space plus 4 n space plus space 2 space plus space 4 close square brackets over denominator 4 end fraction begin inline style blank end style
begin inline style S subscript n end style begin inline style space end style begin inline style equals end style begin inline style space end style begin inline style fraction numerator n left parenthesis n plus 1 right parenthesis open square brackets n squared space plus space 5 n space plus space 6 close square brackets over denominator 4 end fraction end style begin inline style blank end style
begin inline style F end style begin inline style a end style begin inline style c end style begin inline style t end style begin inline style o end style begin inline style r end style begin inline style i end style begin inline style z end style begin inline style i end style begin inline style n end style begin inline style g end style begin inline style space end style begin inline style b end style begin inline style y end style begin inline style space end style begin inline style s end style begin inline style p end style begin inline style l end style begin inline style i end style begin inline style t end style begin inline style t end style begin inline style i end style begin inline style n end style begin inline style g end style begin inline style space end style begin inline style o end style begin inline style f end style begin inline style space end style begin inline style m end style begin inline style i end style begin inline style d end style begin inline style d end style begin inline style l end style begin inline style e end style begin inline style space end style begin inline style t end style begin inline style e end style begin inline style r end style begin inline style m end style begin inline style s end style begin inline style space end style begin inline style o end style begin inline style f end style begin inline style space end style begin inline style n squared end style begin inline style space end style begin inline style plus end style begin inline style space end style begin inline style 5 end style begin inline style n end style begin inline style space end style begin inline style plus end style begin inline style space end style begin inline style 6 end style begin inline style blank end style
begin inline style S subscript n end style begin inline style space end style begin inline style equals end style begin inline style space end style begin inline style fraction numerator n left parenthesis n plus 1 right parenthesis open square brackets n squared space plus space 3 n space plus space 2 n space plus space 6 close square brackets over denominator 4 end fraction end style begin inline style blank end style begin inline style blank end style
begin inline style S subscript n end style begin inline style space end style begin inline style equals end style begin inline style space end style begin inline style fraction numerator n left parenthesis n plus 1 right parenthesis open square brackets n left parenthesis n plus 3 right parenthesis space plus space 2 left parenthesis n space plus space 3 right parenthesis close square brackets over denominator 4 end fraction end style begin inline style blank end style
begin inline style S subscript n end style begin inline style space end style begin inline style equals end style begin inline style space end style begin inline style fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis n plus 2 right parenthesis left parenthesis n space plus space 3 right parenthesis over denominator 4 end fraction end style begin inline style blank end style begin inline style blank end style

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