The sum of the first 3 terms of a geometric sequence is 14. If the common ratio is −3, find the sum of the…

Question

The sum of the first 3 terms of a geometric sequence is 14. If the common ratio is −3, find the sum of the first 5 terms.

ScanSolve AI Tutor · Accepted Answer

Hili ni swali kuhusu mfululizo wa kijiometri (geometric sequence). 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 $S_3 = 14$ na $r = -3$ .
$14 = \frac{a((-3)^3 - 1)}{-3 - 1}$
Rahisisha na utatue kwa $a$ .
$14 = \frac{a(-27 - 1)}{-4}$ $14 = \frac{a(-28)}{-4}$ $14 = 7a$ $a = \frac{14}{7}$ $a = 2$
Sasa tumia thamani ya $a$ uliyopata na $r=-3$ ili kupata jumla ya vigezo 5 vya kwanza ( $S_5$ ).
$S_5 = \frac{a(r^5 - 1)}{r - 1}$ $S_5 = \frac{2((-3)^5 - 1)}{-3 - 1}$
Fanya hesabu.
$S_5 = \frac{2(-243 - 1)}{-4}$ $S_5 = \frac{2(-244)}{-4}$ $S_5 = \frac{-488}{-4}$ $S_5 = 122$
Jumla ya vigezo 5 vya kwanza ni:
$122$

The sum of the first 3 terms of a geometric sequence is 14. If the common ratio is -3, find the sum of the first 5 terms.