Permutations and Combinations

Find the least value of n for which the sum 1 + 3 + 3 2 + ...... to n terms is greater than 7000. - Wired Faculty

Question

Find the least value of n for which the sum 1 + 3 + 3²+ ...... to n terms is greater than 7000.

Solution

Let $space space straight S subscript straight n space equals space 1 plus 3 plus 3 squared plus...... space to space straight n space terms$
Here, a = 1, r = 3
$straight S subscript straight n. space space space space fraction numerator straight a left parenthesis straight r to the power of straight n minus 1 right parenthesis over denominator straight r minus 1 end fraction space equals space fraction numerator 1 left parenthesis 3 to the power of straight n minus 1 right parenthesis over denominator 3 minus 1 end fraction space equals space fraction numerator 3 to the power of straight n minus 1 over denominator 2 end fraction$
Since $straight S subscript straight n greater than 7000$

∴ $fraction numerator 3 to the power of straight n minus 1 over denominator 2 end fraction greater than 7000$ $rightwards double arrow space space space space space space space space space space space space space space space space space 3 to the power of straight n minus 1 space greater than space 14000$ $rightwards double arrow space space space space space space space space space space space 3 to the power of straight n greater than 14001$
$rightwards double arrow$ $3 to the power of straight n space greater than space 19683 space greater than space 14001 space greater than space 6561$
$rightwards double arrow$ $3 to the power of straight n greater than space 3 to the power of 9 greater than 14001 space greater than space 3 to the power of 8$
Hence, the least value of n is 9.

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