The 1st term of a G.P. is 4. If the common ratio is 2, find the greatest number of terms that will give a sum less than 40.

Tunapewa:
Neno la kwanza, $a = 4$
Uwiano wa kawaida, $r = 2$
Jumla ya maneno $n$, $S_n < 40$

Fomula ya jumla ya maneno $n$ ya GP wakati $r > 1$ ni:

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

$$S_n = \frac{4(2^n - 1)}{2 - 1}$$ $$S_n = \frac{4(2^n - 1)}{1}$$ $$S_n = 4(2^n - 1)$$

Gawanya pande zote mbili kwa 4:

$$\frac{4(2^n - 1)}{4} < \frac{40}{4}$$ $$2^n - 1 < 10$$

Ongeza 1 kwa pande zote mbili:

$$2^n - 1 + 1 < 10 + 1$$ $$2^n < 11$$

Tujaribu thamani za $n$:
Kwa $n = 1$, $2^1 = 2$ (ambayo ni chini ya 11)
Kwa $n = 2$, $2^2 = 4$ (ambayo ni chini ya 11)
Kwa $n = 3$, $2^3 = 8$ (ambayo ni chini ya 11)
Kwa $n = 4$, $2^4 = 16$ (ambayo SI chini ya 11)

Kwa hiyo, thamani kubwa zaidi ya $n$ ambayo inakidhi $2^n < 11$ ni $n = 3$.

Nambari kubwa zaidi ya maneno ambayo itatoa jumla chini ya 40 ni 3.

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