Question

EXERCISE 5.4 (Optional)
1. Which term of the AP: 121, 117, 113...., is
its first negative term?
[Hint: Find n for a <0]
2. The sum of the third and the seventh terms
of an AP is 6 and their product is 8. Find
the sum of first sixteen terms of the AP.
3. A ladder has rungs 25 cm apart.
(see Fig. 5.7). The rungs decrease
uniformly in length from 45 cm at the
bottom to 25 cm at the top. If the top and
the bottom rungs are 25m apart, what is
the length of the wood required for the
rungs?
25 cm
45 cm
Fig. 5.7
[Hint: Number of rungs =
4. The houses of a row are numbered consecutively from 1 to 49. Show that there is a value
of x such that the sum of the numbers of the houses preceding the house numbered xis
equal to the sum of the numbers of the houses following it. Find this value of x.
[Hint: S-=S-S)
5. A small terrace at a football ground comprises of 15 steps each of which is 50 m long and
built of solid concrete.
Each step has a rise of mand a tread of m. (see Fig. 5.8). Calculate the total volume
of concrete required to build the terrace.
1
[Hint : Volume of concrete required to build the first step=x - x 50 m]
--50 m
Fig. 5.8​

Answer 1

Answer:

The first negetive term of the given A.P is it's 32nd term, (n=32)

Answer 2

$\orange{\boxed{\pink{\underline{\red{\mathrm{Solution:-}}}}}}$

Q.1)↓

a=121

d=-4[117-121]

1ˢᵗ negetive term, -- an<0

an=a+(n-1)d<0

→121+(n-1)×-4<0

→(n-1)×-4<-121

→(n-1)<$\frac{-121}{-4}$

=>(n-1)<30.25

n>=30.25+1

=>n>31.25

→n is greater than 31.25 which means 32ⁿᵈ term is 1st Negetive Term✓

Q.2)↓

a3+a7=6

a3×a7=8

a16=?

expanding a3+a7

→a+2d+a+6d=6

=>2a+8d=6 -->÷by 2

→a+4d=3

→a=3-4d--(1)

Expanding a3×a7

→(a+2d)(a+6d)=8

→Use (1)

=>(3-4d+2d)(3-4d+6d)=8

→(3-2d)×(3+2d)=8

a²-b²=(a+b)(a-b) using this↓

→(3)²-(2d)²=8

→9-4d²=8

-4d²=8-9

-4d²=-1 [LHS(minus)&RHS(minus) cancelled]

d²=$\frac{1}{4}$

d=

$\sqrt{ \frac{1}{4} }$

→d=$\frac{1}{2}$

→a=3-$\frac{\cancel{4}}{\cancel{2}}$

→a=3-2=1

→$\boxed{a=1}$

S16=$\frac{16}{2}$×(2×1+(16-1)×½)

S16=8×(2+$\frac{15}{2}$ )

Cross Multiply 2+$\frac{15}{2}$

S16=8×$\frac{19}{2}$

S16=4×19

$\orange{\boxed{\pink{\underline{\red{\mathrm{S=76}}}}}}$

$\mathcal{BE \: BRAINLY}$