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8. Find the 8th term and the sum of the first 15 terms of the AP: $-7, -4, -1, \dots$

Identify the first term ( $a$ ) and the common difference ( $d$ ).
The first term is $a = -7$ .
The common difference is $d = -4 - (-7) = -4 + 7 = 3$ .
Find the 8th term ( $T_8$ ) using the formula $T_n = a + (n-1)d$ .
$T_8 = -7 + (8-1)3$ $T_8 = -7 + (7)3$ $T_8 = -7 + 21$ $T_8 = 14$
Find the sum of the first 15 terms ( $S_{15}$ ) using the formula $S_n = \frac{n}{2}[2a + (n-1)d]$ .
$S_{15} = \frac{15}{2}[2(-7) + (15-1)3]$ $S_{15} = \frac{15}{2}[-14 + (14)3]$ $S_{15} = \frac{15}{2}[-14 + 42]$ $S_{15} = \frac{15}{2}[28]$ $S_{15} = 15 \times 14$ $S_{15} = 210$
The 8th term is $\boxed{14}$ and the sum of the first 15 terms is $\boxed{210}$ .

9. An arc of a circle, with a radius of 10cm, subtends an angle of $80^\circ$ at the centre of the circle. Find the length of the arc.
Identify the radius ( $r$ ) and the angle ( $\theta$ ).
The radius $r = 10$ cm.
The angle $\theta = 80^\circ$ .
Use the formula for arc length $L = \frac{\theta}{360^\circ} \times 2\pi r$ .
$L = \frac{80^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 10$ $L = \frac{8}{36} \times 2 \times \frac{22}{7} \times 10$ $L = \frac{2}{9} \times \frac{440}{7}$ $L = \frac{880}{63}$ $L \approx 13.968$
The length of the arc is approximately $\boxed{13.97 \text{ cm}}$ .

10. In a trapezium with an area of $40 \text{cm}^2$ , the ratio of the lengths of the parallel sides is $2:3$ . Find the lengths of the parallel sides if they are separated by 5cm.
Identify the area ( $A$ ), the ratio of parallel sides, and the height ( $h$ ).
The area $A = 40 \text{cm}^2$ .
The ratio of parallel sides $a:b = 2:3$ .
The height $h = 5$ cm.
Let the parallel sides be $2x$ and $3x$.
Use the formula for the area of a trapezium $A = \frac{1}{2}(a+b)h$ .
$40 = \frac{1}{2}(2x + 3x)5$ $40 = \frac{1}{2}(5x)5$ $40 = \frac{25x}{2}$
Solve for $x$ .
$40 \times 2 = 25x$ $80 = 25x$ $x = \frac{80}{25}$ $x = \frac{16}{5}$ $x = 3.2$
Find the lengths of the parallel sides.
The first side $a = 2x = 2 \times 3.2 = 6.4$ cm.
The second side $b = 3x = 3 \times 3.2 = 9.6$ cm.

The lengths of the parallel sides are $\boxed{6.4 \text{ cm}}$ and $\boxed{9.6 \text{ cm}}$ .

Can you use English please

Step-by-step solution

Identify the first term ( $a$ ) and the common difference ( $d$ ).

Find the 8th term ( $T_8$ ) using the formula $T_n = a + (n-1)d$ .

Find the sum of the first 15 terms ( $S_{15}$ ) using the formula $S_n = \frac{n}{2}[2a + (n-1)d]$ .

Identify the radius ( $r$ ) and the angle ( $\theta$ ).

Use the formula for arc length $L = \frac{\theta}{360^\circ} \times 2\pi r$ .

Identify the area ( $A$ ), the ratio of parallel sides, and the height ( $h$ ).

Let the parallel sides be $2x$ and $3x$.

Solve for $x$ .

Find the lengths of the parallel sides.

Still stuck on this one?

More English Questions

Can you use English please

Step-by-step solution

Identify the first term (aaa) and the common difference (ddd).

Find the 8th term (T8T_8T8​) using the formula Tn=a+(n−1)dT_n = a + (n-1)dTn​=a+(n−1)d.

Find the sum of the first 15 terms (S15S_{15}S15​) using the formula Sn=n2[2a+(n−1)d]S_n = \frac{n}{2}[2a + (n-1)d]Sn​=2n​[2a+(n−1)d].

Identify the radius (rrr) and the angle (θ\thetaθ).

Use the formula for arc length L=θ360∘×2πrL = \frac{\theta}{360^\circ} \times 2\pi rL=360∘θ​×2πr.

Identify the area (AAA), the ratio of parallel sides, and the height (hhh).

Let the parallel sides be $2x$ and $3x$.

Solve for xxx.

Find the lengths of the parallel sides.

Still stuck on this one?

More English Questions

Identify the first term ( $a$ ) and the common difference ( $d$ ).

Find the 8th term ( $T_8$ ) using the formula $T_n = a + (n-1)d$ .

Find the sum of the first 15 terms ( $S_{15}$ ) using the formula $S_n = \frac{n}{2}[2a + (n-1)d]$ .

Identify the radius ( $r$ ) and the angle ( $\theta$ ).

Use the formula for arc length $L = \frac{\theta}{360^\circ} \times 2\pi r$ .

Identify the area ( $A$ ), the ratio of parallel sides, and the height ( $h$ ).

Solve for $x$ .