Define the series T_n= _ k=1 ^ n (3k+1 ) ( 12 )^ k-1

MATHEMATICS · Subjective · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series

Problem

Define the series Tn=∑k=1n(3k+1)(12)k−1.T_n=\sum_{k=1}^{n} \bigl(3k+1\bigr)\left(\frac12\right)^{k-1}. (i) Rewrite TnT_n as a combination of two separate summations, each of which can be evaluated using a standard formula. (ii) Hence obtain a simplified closed-form expression for TnT_n in terms of nn. (iii) Using your result from part (ii), calculate T10T_{10} correct to two decimal places.

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