The sum of the first 6 terms of a geometric series is 264. If the common ratio is 3, find the first term.

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ScanSolve AI Tutor · Accepted Answer

Hili ni swali lingine kuhusu mfululizo wa kijiometri. Tutatumia formula ya jumla ya vigezo vya mfululizo wa kijiometri.

Formula ya jumla ya $n$ vigezo vya kwanza vya mfululizo wa kijiometri ni:

S_n = \frac{a(r^n - 1)}{r - 1}

ambapo $a$ ni kigezo cha kwanza, $r$ ni uwiano wa kawaida (common ratio), na $n$ ni idadi ya vigezo.

Tumia taarifa uliyopewa kupata kigezo cha kwanza ( $a$ ).
Tunajua $S_4 = 26$ na $r = 3$ .
$26 = \frac{a(3^4 - 1)}{3 - 1}$
Rahisisha na utatue kwa $a$ .
$26 = \frac{a(81 - 1)}{2}$ $26 = \frac{a(80)}{2}$ $26 = 40a$ $a = \frac{26}{40}$ $a = \frac{13}{20}$
Sasa tumia thamani ya $a$ uliyopata na $r=3$ ili kupata jumla ya vigezo 6 vya kwanza ( $S_6$ ).
$S_6 = \frac{a(r^6 - 1)}{r - 1}$ $S_6 = \frac{\frac{13}{20}(3^6 - 1)}{3 - 1}$
Fanya hesabu.
$S_6 = \frac{\frac{13}{20}(729 - 1)}{2}$ $S_6 = \frac{\frac{13}{20}(728)}{2}$ $S_6 = \frac{13 \times 728}{20 \times 2}$ $S_6 = \frac{13 \times 728}{40}$ $S_6 = \frac{9464}{40}$
Rahisisha jibu.
$S_6 = \frac{2366}{10}$ $S_6 = \frac{1183}{5}$
Jumla ya vigezo 6 vya kwanza ni:
$\frac{1183}{5}$

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