Find the eighth term and the sum of the first 15 terms of the AP. −7, −4, −1, .........

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8. Pata neno la nane na jumla ya maneno 15 ya kwanza ya AP: $-7, -4, -1, \dots$

Tambua neno la kwanza ( $a$ ) na tofauti ya kawaida ( $d$ ).
Neno la kwanza $a = -7$ .
Tofauti ya kawaida $d = -4 - (-7) = -4 + 7 = 3$ .
Pata neno la nane ( $T_8$ ) kwa kutumia fomula $T_n = a + (n-1)d$ .
$T_8 = -7 + (8-1)3$ $T_8 = -7 + (7)3$ $T_8 = -7 + 21$ $T_8 = 14$
Pata jumla ya maneno 15 ya kwanza ( $S_{15}$ ) kwa kutumia fomula $S_n = \frac{n}{2}[2a + (n-1)d]$ .
$S_{15} = \frac{15}{2}[2(-7) + (15-1)3]$ $S_{15} = \frac{15}{2}[-14 + (14)3]$ $S_{15} = \frac{15}{2}[-14 + 42]$ $S_{15} = \frac{15}{2}[28]$ $S_{15} = 15 \times 14$ $S_{15} = 210$
Neno la nane ni $\boxed{14}$ na jumla ya maneno 15 ya kwanza ni $\boxed{210}$ .

9. Tao la duara, lenye rediasi 10cm, linatengeneza pembe ya $80^\circ$ katikati ya duara. Pata urefu wa tao.
Tambua rediasi ( $r$ ) na pembe ( $\theta$ ).
Rediasi $r = 10$ cm.
Pembe $\theta = 80^\circ$ .
Tumia fomula ya urefu wa tao $L = \frac{\theta}{360^\circ} \times 2\pi r$ .
$L = \frac{80^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 10$ $L = \frac{8}{36} \times 2 \times \frac{22}{7} \times 10$ $L = \frac{2}{9} \times \frac{440}{7}$ $L = \frac{880}{63}$ $L \approx 13.968$
Urefu wa tao ni takriban $\boxed{13.97 \text{ cm}}$ .

10. Katika trapezi yenye eneo la $40 \text{cm}^2$ , uwiano wa urefu wa pande sambamba ni $2:3$ . Pata urefu wa pande sambamba ikiwa zimetenganishwa kwa 5cm.
Tambua eneo ( $A$ ), uwiano wa pande sambamba, na urefu ( $h$ ).
Eneo $A = 40 \text{cm}^2$ .
Uwiano wa pande sambamba $a:b = 2:3$ .
Urefu $h = 5$ cm.
Wacha pande sambamba ziwe $2x$ na $3x$.
Tumia fomula ya eneo la trapezi $A = \frac{1}{2}(a+b)h$ .
$40 = \frac{1}{2}(2x + 3x)5$ $40 = \frac{1}{2}(5x)5$ $40 = \frac{25x}{2}$
Tatua kwa $x$ .
$40 \times 2 = 25x$ $80 = 25x$ $x = \frac{80}{25}$ $x = \frac{16}{5}$ $x = 3.2$
Pata urefu wa pande sambamba.
Upande wa kwanza $a = 2x = 2 \times 3.2 = 6.4$ cm.
Upande wa pili $b = 3x = 3 \times 3.2 = 9.6$ cm.

Urefu wa pande sambamba ni $\boxed{6.4 \text{ cm}}$ na $\boxed{9.6 \text{ cm}}$ .

Find the eighth term and the sum of the first 15 terms of the AP. -7, -4, -1, .........

Step-by-step solution

Tambua neno la kwanza ( $a$ ) na tofauti ya kawaida ( $d$ ).

Pata neno la nane ( $T_8$ ) kwa kutumia fomula $T_n = a + (n-1)d$ .

Pata jumla ya maneno 15 ya kwanza ( $S_{15}$ ) kwa kutumia fomula $S_n = \frac{n}{2}[2a + (n-1)d]$ .

Tambua rediasi ( $r$ ) na pembe ( $\theta$ ).

Tumia fomula ya urefu wa tao $L = \frac{\theta}{360^\circ} \times 2\pi r$ .

Tambua eneo ( $A$ ), uwiano wa pande sambamba, na urefu ( $h$ ).

Wacha pande sambamba ziwe $2x$ na $3x$.

Tatua kwa $x$ .

Pata urefu wa pande sambamba.

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Find the eighth term and the sum of the first 15 terms of the AP. -7, -4, -1, .........

Step-by-step solution

Tambua neno la kwanza (aaa) na tofauti ya kawaida (ddd).

Pata neno la nane (T8T_8T8​) kwa kutumia fomula Tn=a+(n−1)dT_n = a + (n-1)dTn​=a+(n−1)d.

Pata jumla ya maneno 15 ya kwanza (S15S_{15}S15​) kwa kutumia fomula Sn=n2[2a+(n−1)d]S_n = \frac{n}{2}[2a + (n-1)d]Sn​=2n​[2a+(n−1)d].

Tambua rediasi (rrr) na pembe (θ\thetaθ).

Tumia fomula ya urefu wa tao L=θ360∘×2πrL = \frac{\theta}{360^\circ} \times 2\pi rL=360∘θ​×2πr.

Tambua eneo (AAA), uwiano wa pande sambamba, na urefu (hhh).