Question

8. The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find the 27th term.

Answer 1

AP ( Arithmetic progression).
An arithmetic progression is a list of numbers a1, a2, a3 ………….. an in which each term is obtained by adding a fixed number to the preceding term except the first term.
This fixed number is called the common difference( d ) of the AP. Common difference of an AP will be the difference between any two consecutive terms.

a2= a1+d
a3= a2+d
a4= a3+d
……..
an= an-1+d ………

Each of the numbers in the list is called a term .

Method to find the common difference :
d = a2 - a1 or a3 - a2 or a4 - a3...

General form of an AP.:
a, a+d, a+2d, a+3d…….

Here a is the first term and d is common difference.

General term or nth term of A.P
The general term or nth term of A.P is given by an = a + (n – 1)d, where a is the first term, d is the common difference and n is the number of term.

SOLUTION :
GIVEN : a10 = 41, a18 = 73

an = a + (n – 1)d
a10 = a+(n -1)d
41 = a + (10 -1)d
41 = a+9d
a +9d = 41………… (1)

a18 = a+(n-1)d
73 = a+ (18-1)d
73 = a+17d
a +17d = 73……………(2)

ON SUBTRACTING eq 1 FROM 2, WE GET
a +17d = 73
a + 9d = 41
(-) (-) (-)
-------------------
8d = 32
d = 32/8
d = 4

Put d = 4 in eq 1
a +9d = 41
41 = a + 9(4)
41 = a + 36
41 - 36 = a
a = 5

a27 = a +(n-1)4
a27 = 5 + (27 -1)4
a27 = 5 + 26 (4)
a27 = 5 + 104
a27= 109

a27 or t27 = 109

Hence, the 27th term is 109

HOPE THIS WILL HELP YOU….

Answer 2

let first term be a , common difference be d.
given, a₁₀ = 41

a₁₀ = a + (10 - 1)d

41 = a + 9d----------( 1 ).

a₁₈ = 73

a₁₈ = a + (18 - 1)d

73 = a + 17d------------( 2 ).

From--------( 1 ) & --------( 2 )

a + 17d = 73
a + 9d = 41
------------------
8d = 32

d = 4 [put in ------( 1 )]

a + 9(4) = 41

a = 41 - 36

a = 5 , d = 4

now ,
a₂₇ = a + (27 - 1)d
a₂₇ = 5 + 26(4)
a₂₇ = 5 + 104
a₂₇ = 109