Question

14. The angles of a triangle are in AP. If the
greatest angle equals to the sum of the
other two, then find the angles. Also, find
these angles are multiple of which angle.

15. A sum of 1000 is invested at 8% simple
interest per annum. Calculate the interest
at the end of 1, 2, 3 ... years. Is the sequence of interests an AP? Find the
interest at the end of 30 years?​

Answer 1

Step-by-step explanation:

1.(a-d),(a),& (a+d).

(a-d)+(a)+ (a+d) = 180°

3a = 180°

a= 180 /3 = 60 °

The 3 angles are 60° -d , 60° and 60°+ d.

ATQ

Greatest angle = sum of two smaller angles

60° + d = 60° -d + 60°

60° + d = 120° -d

d+d = 120° -60°

2d = 60°

d = 60° /2= 30°

d = 30°

Required angles of a triangle are (60° -30°) , (60°) and (60° + 30°)=  30° , 60°, 90°.

Hence, the required angles of a triangle are 30° , 60°, 90°.

These angles are the multiple of 30°.

2.Given,

principal=1000

rate of interest=8%

number of years=1 year

SI=Pnr/100=1000x1x8/100=rs.80

when n=2 years,SI=1000x2x8/100=rs.160

when n=3years,SI=1000x3x8/100=rs.240

the sequence is

80,160,240,..........

this is in AP as a,first term=80,d,common difference=80

interest at the end of 30 years,t30=a+(n-1)d

=80+(30-1)80

=80+29(80)

=80+2320

=2400

Answer 2

14.

Give:

• The angles of a triangle are in AP

• Also - the greatest angle equals the sum of the  other two

To be found:

Angles and also find these angles are multiple of which angle

Let the three angles be '(a - d)', 'a' and '(a + d)'

By the angle sum property of triangle we know- the sum of all three angles of triangle = 180°

So,

⇒ (a - d) + a + (a + d) = 180

⇒ a - d + a + a + d = 180

⇒ a + a + a + d - d = 180

⇒ 3a = 180

⇒ a = 180 ÷ 3

⇒ a = 60°

Now,

According to the question,

• The greatest angle is equal to the sum of the other two angles.

⇒ 60° + d = 60° + (60° - d)

⇒ 60° + d = 120° - d

⇒ d + d = 120° - 60°

⇒ 2d = 60°

⇒ d = 60° ÷ 2

⇒ d = 30°

So,

Value of a = 60° and d = 30°

Hence,

The angles are

a = 60°

a - d = 60° - 30° = 30°

a + d = 60° + 30° = 90°

We can observe that all angles are multiples of 30°.

- - - - -

15.

Given:

Principal (P)= Rs.1000

Rate of Interest (R%) = 8% per annum

To find:

Calculating the interest at the end of 1, 2, 3, ..... years.

So,

We can use the formula of Siple Interest(S.I) here,

$\boxed{S.I = \frac{P \times R \times T}{100}}$

Now, at the end of 1 year, we get,

$S.I = \frac{10 \not0 \not0 \times 8 \times 1}{1\not0 \not0} =\boxed{ Rs.80}$

At the end of 2 years, we get,

$S.I = \frac{10 \not0 \not0 \times 8 \times 2}{1\not0 \not0} =\boxed{ Rs.160}$

At the end of 2 years, we get,

$S.I = \frac{10 \not0 \not0 \times 8 \times 3}{1\not0 \not0} =\boxed{ Rs.240}$

Looking all clearly we get that Rs.80 is more than the previous one.

Therefore,

80, 160, 240..... is the sequence of interest. So it is an AP.

We get a = Rs.80,

And common difference (d) = a₂ - a₁

= 160 - 80 = Rs.80

Now, by using the formula for nth term,

$\boxed{a_{n} = a+(n-1)d}$

So,

a₃₀ = 80 + (30 -1)(80)

     = 80 + 29(80)

     = 80 + 2320

     = Rs.2400

Hence,

The interest (S.I) at the end of 30 years is Rs.2400