If the first term of a geometric series is 4 and the common ratio is 2, find the sum of the first 8 terms.

Question

ScanSolve AI Tutor · Accepted Answer

Hili ni swali kuhusu mfululizo wa kijiometri (geometric series). 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 $S_9 = 24$ na $r = 2$ .
$24 = \frac{a(2^9 - 1)}{2 - 1}$
Rahisisha na utatue kwa $a$ .
$24 = \frac{a(512 - 1)}{1}$ $24 = a(511)$ $a = \frac{24}{511}$
Sasa tumia thamani ya $a$ uliyopata na $r=2$ ili kupata jumla ya vigezo 8 vya kwanza ( $S_8$ ).
$S_8 = \frac{a(r^8 - 1)}{r - 1}$ $S_8 = \frac{\frac{24}{511}(2^8 - 1)}{2 - 1}$
Fanya hesabu.
$S_8 = \frac{\frac{24}{511}(256 - 1)}{1}$ $S_8 = \frac{24}{511}(255)$ $S_8 = \frac{24 \times 255}{511}$ $S_8 = \frac{6120}{511}$
Jumla ya vigezo 8 vya kwanza ni:
$\frac{6120}{511}$

Answer