The 1st, 5th and 7th terms of an AP form the first three consecutive terms of a GP. The common difference of AP is d and the common ratio of GP is r. (a) Find the values of d and r (b) Find the sum of the first 15 terms of the AP

Acha nambari ya kwanza ya AP iwe $a$.
Tofauti ya kawaida ya AP ni $d$.
Nambari ya $n^{\text{th}}$ ya AP inatolewa na $T_n = a + (n-1)d$.
Hivyo, nambari za AP zinazohusika ni:
$T_1 = a$
$T_5 = a + (5-1)d = a + 4d$
$T_7 = a + (7-1)d = a + 6d$

Nambari hizi $T_1, T_5, T_7$ huunda GP na uwiano wa kawaida $r=32$.
Kwa GP, uwiano wa nambari mfululizo ni sawa na uwiano wa kawaida.

Kutoka Equation 1:

Kutoka Equation 2:

Panga upya ili kupata $a$ na $d$:

Sasa badilisha $a = \frac{4d}{31}$ katika equation hii:

Tumia $a = \frac{4d}{31}$ na $d=0$:

Kwa hivyo, $d=0$ na $r=32$ (imetolewa).

a) Majibu:
$\boxed{d = 0}$
$\boxed{r = 32}$

b) Tafuta jumla ya nambari 15 za kwanza za AP.

Jumla ya nambari $n$ za kwanza za AP inatolewa na formula:

Tunahitaji jumla ya nambari 15 za kwanza, hivyo $n=15$.
Kutoka sehemu (a), tuna $a=0$ na $d=0$.

b) Jibu:
$\boxed{S_{15} = 0}$