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CBSE Class 12-science Answered

If the n terms a1, a2, . . . , an are in arithmetic progression with increment r, then the difference between the mean of their squares and the square of their mean is
Asked by jajravishan | 25 Feb, 2020, 06:41: PM
answered-by-expert Expert Answer
The terms a1, a2, a3, ..., an are in AP with common difference r
Therefore, a2 = a1 + r, a3 = a1 + 2r, ... an = a1 + (n-1)r
D i f f e r e n c e space b e t w e e n space t h e space m e a n space o f space t h e i r space s q u a r e s space a n d space t h e space s q u a r e space o f space t h e i r space m e a n equals open vertical bar fraction numerator a subscript 1 squared plus a subscript 2 squared plus a subscript 3 squared plus.... plus a subscript n squared over denominator n end fraction minus fraction numerator a subscript 1 plus a subscript 2 plus a subscript 3 plus... plus a subscript n over denominator n end fraction close vertical bar equals open vertical bar fraction numerator a subscript 1 squared plus open parentheses a subscript 1 plus r close parentheses squared plus open parentheses a subscript 1 plus 2 r close parentheses squared plus.... plus open square brackets a subscript 1 plus open parentheses n minus 1 close parentheses r close square brackets squared over denominator n end fraction minus fraction numerator a subscript 1 plus a subscript 1 plus r plus a subscript 1 plus 2 r plus.... plus a subscript 1 plus open parentheses n minus 1 close parentheses r over denominator n end fraction close vertical bar equals open vertical bar fraction numerator a subscript 1 squared plus open parentheses a subscript 1 squared plus r squared plus 2 r close parentheses plus open parentheses a subscript 1 squared plus 4 r squared plus 4 r close parentheses plus.... plus open parentheses a subscript 1 squared plus open parentheses n minus 1 close parentheses squared r squared plus 2 left parenthesis n minus 1 right parenthesis r close parentheses over denominator n end fraction minus fraction numerator n a subscript 1 plus r open square brackets 1 plus 2 plus... plus left parenthesis n minus 1 right parenthesis close square brackets over denominator n end fraction close vertical bar equals open vertical bar fraction numerator n a subscript 1 squared plus r squared open square brackets 1 plus 4 plus 9 plus... plus left parenthesis n minus 1 right parenthesis squared close square brackets plus 2 r open square brackets 1 plus 2 plus 3 plus... plus open parentheses n minus 1 close parentheses close square brackets over denominator n end fraction minus fraction numerator n a subscript 1 plus r open square brackets 1 plus 2 plus... plus left parenthesis n minus 1 right parenthesis close square brackets over denominator n end fraction close vertical bar equals open vertical bar a subscript 1 squared plus fraction numerator r squared open parentheses n minus 1 close parentheses open parentheses n close parentheses open parentheses 2 n minus 1 close parentheses over denominator n end fraction plus fraction numerator 2 r open parentheses n minus 1 close parentheses n over denominator 2 n end fraction minus a subscript 1 minus fraction numerator r open parentheses n minus 1 close parentheses n over denominator n end fraction close vertical bar equals open vertical bar a subscript 1 squared plus r squared open parentheses n minus 1 close parentheses open parentheses 2 n minus 1 close parentheses minus a subscript 1 close vertical bar equals open vertical bar a subscript 1 squared minus a subscript 1 plus r squared open parentheses 2 n squared minus 3 n plus 1 close parentheses close vertical bar
Answered by Renu Varma | 27 Feb, 2020, 10:13: AM

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