Question

The 23rd term and 37th term of an AP are 54, 96 Find 30th term​

Answer 1

EXPLANATION.

23rd term of an A.P. = 54.

37th term of an A.P. = 96.

As we know that,

General term of an A.P.

⇒ Tₙ = a + (n - 1)d.

⇒ T₂₃ = a + (23 - 1)d.

⇒ T₂₃ = a + 22d.

⇒ a + 22d = 54. ⇒ (1)

⇒ T₃₇ = a + (37 - 1)d.

⇒ T₃₇ = a + 36d.

⇒ a + 36d = 96. ⇒ (2).

From equation (1) & (2), we get.

⇒ a + 22d = 54.

⇒ a + 36d = 96.

We get.

⇒ - 14d = - 42.

⇒ 14d = 42.

⇒ d = 42/14.

⇒ d = 3.

Put the value of d = 3 in equation (1), we get.

⇒ a + 22d = 54.

⇒ a + 22(3) = 54.

⇒ a + 66 = 54.

⇒ a = 54 - 66.

⇒ a = -12.

To find 30th term of an A.P.

⇒ T₃₀ = a + (30 - 1)d.

⇒ T₃₀ = a + 29d.

⇒ T₃₀ = -12 + 29(3).

⇒ T₃₀ = -12 + 87.

⇒ T₃₀ = 75.

                                                                                                                       

MORE INFORMATION.

Supposition of terms in A.P.

(1) = Three terms as : a - d, a, a + d.

(2) = Four terms as : a - 3d, a - d, a + d, a + 3d.

(3) = Five terms as : a - 2d, a - d, a, a + d, a + 2d.

Answer 2

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❦ANSWER彡

GIVEN :-

$23rd \: \: term = 54 \\ 37th \: term = 96$

‣30th term = ?

➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖

1) 23rd term

➣T(23) = a+(23-1) d = 54

➣T(23) = a+22d = 54

➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖

2) 37th term

➣T(37) = a+ (37-1)d = 96

➣T(37) = a+36d = 96

➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖

‣NOW HAVE TO FIND 30th TERM :-

First equation = T(23) = a+22d = 54

Second equation = T(37) = a+36d = 96

➣ a + 22d = 54

➣ a + 36d = 96

‣ -14d = -42

‣ d = 3

➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖

NOW, PUT d = 3

➣ a + 22d = 54

➣a+22(3)= 54

➣a+66=54

➣a = -12

➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖➖

30th TERM :-

‣T(30) = a+ (30-1) d

‣T(30) = -12+ (30-1) (3)

‣T(30) = -12+ (29) (3)

‣T(30) = -12 + 87

‣ T(30) = 75

SO, THE VALUE OF 30th TERM = 75

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