Solve yhis

Asked by jeeveshjain87 | 23rd Sep, 2021, 11:39: PM

Expert Answer:

Let S equals sum from r equals 1 to n of open parentheses sum from k equals 1 to r of k close parentheses open square brackets log subscript 1 divided by 2 end subscript square root of 4 x minus x squared end root close square brackets to the power of r equals sum from r equals 1 to n of open square brackets fraction numerator r open parentheses r plus 1 close parentheses over denominator 2 end fraction close square brackets open square brackets log subscript 1 divided by 2 end subscript square root of 4 x minus x squared end root close square brackets to the power of r
Take A equals open square brackets log subscript 1 divided by 2 end subscript square root of 4 x minus x squared end root close square brackets
rightwards double arrow straight S equals 1 half open square brackets 1 cross times 2 straight A plus 2 cross times 3 straight A squared plus 3 cross times 4 straight A cubed plus... plus straight n open parentheses straight n plus 1 close parentheses straight A to the power of straight n close square brackets

Suppose space straight S subscript 1 equals 1 cross times 2 straight A plus 2 cross times 3 straight A squared plus 3 cross times 4 straight A cubed plus... plus straight n open parentheses straight n plus 1 close parentheses straight A to the power of straight n
rightwards double arrow straight S equals 1 half straight S subscript 1
Now comma space straight S subscript 1 equals 1 cross times 2 straight A plus 2 cross times 3 straight A squared plus 3 cross times 4 straight A cubed plus... plus straight n open parentheses straight n plus 1 close parentheses straight A to the power of straight n
rightwards double arrow straight S subscript 1 equals 2 straight A plus 6 straight A squared plus 12 straight A cubed plus... plus straight n open parentheses straight n plus 1 close parentheses straight A to the power of straight n space space space... space left parenthesis straight i right parenthesis
rightwards double arrow AS subscript 1 equals 0 straight A plus 2 straight A squared plus 6 straight A cubed plus 12 straight A to the power of 4 plus... plus open parentheses straight n minus 1 close parentheses nA to the power of straight n plus straight n open parentheses straight n plus 1 close parentheses straight A to the power of straight n plus 1 end exponent space space space... space left parenthesis ii right parenthesis
Subtracting space left parenthesis ii right parenthesis space from space left parenthesis straight i right parenthesis comma space we space get
open parentheses 1 minus straight A close parentheses straight S subscript 1 equals 2 straight A plus 4 straight A squared plus 6 straight A cubed plus... plus 2 nA to the power of straight n minus straight n open parentheses straight n plus 1 close parentheses straight A to the power of straight n plus 1 end exponent
rightwards double arrow open parentheses 1 minus straight A close parentheses straight S subscript 1 equals 2 open square brackets straight A plus 2 straight A squared plus 3 straight A cubed plus... plus nA to the power of straight n close square brackets minus straight n open parentheses straight n plus 1 close parentheses straight A to the power of straight n plus 1 end exponent
rightwards double arrow open parentheses 1 minus straight A close parentheses straight S subscript 1 equals fraction numerator 2 straight A open parentheses 1 minus straight A to the power of straight n close parentheses over denominator open parentheses 1 minus straight A close parentheses squared end fraction minus fraction numerator 2 nA to the power of straight n plus 1 end exponent over denominator 1 minus straight A end fraction minus straight n open parentheses straight n plus 1 close parentheses straight A to the power of straight n plus 1 end exponent
rightwards double arrow straight S subscript 1 equals fraction numerator 2 straight A open parentheses 1 minus straight A to the power of straight n close parentheses over denominator open parentheses 1 minus straight A close parentheses cubed end fraction minus fraction numerator 2 nA to the power of straight n plus 1 end exponent over denominator open parentheses 1 minus straight A close parentheses squared end fraction minus fraction numerator straight n open parentheses straight n plus 1 close parentheses straight A to the power of straight n plus 1 end exponent over denominator 1 minus straight A end fraction
rightwards double arrow straight S equals fraction numerator straight A open parentheses 1 minus straight A to the power of straight n close parentheses over denominator open parentheses 1 minus straight A close parentheses cubed end fraction minus nA to the power of straight n plus 1 end exponent over open parentheses 1 minus straight A close parentheses squared minus fraction numerator straight n open parentheses straight n plus 1 close parentheses straight A to the power of straight n plus 1 end exponent over denominator 2 open parentheses 1 minus straight A close parentheses end fraction

Answered by Renu Varma | 30th Sep, 2021, 11:42: AM

Queries asked on Sunday & after 7pm from Monday to Saturday will be answered after 12pm the next working day.