Q. 104.3( 16 Votes )

# There are 37 terms in an A.P., the sum of three terms placed exactly at the middle is 225 and the sum of last three terms is 429. Write the A.P.

Answer :

Let first term = a

Common difference = d

Since, A.P. consist of 37 terms, therefor the middle most term is

Thus, three middle most term are t_{18} = 18^{th} term, t_{19} = 19^{th} term,

t_{20} = 20^{th} term

We use n^{th} term of an A.P. formula

t_{n} = a + (n – 1)d

where n = no. of terms

a = first term

d = common difference

t_{n} = n^{th} terms

Thus, on substituting all values we get,

Given, t_{18} + t_{19} + t_{20} = 225

⇒ [a + (18 – 1)d] + [a + (19 – 1)d] + [a + (20 – 1)d] = 225

⇒ [a + 17d] + [a + 18d] + [a + 19d] = 225

⇒ 3a + 54d = 225

Dividing by 3

⇒ a + 18d = 75 …….(1)

Given, sum of last three term is 429

⇒ t_{35} + t_{36} + t_{37} = 429

⇒ [a + (35 – 1)d] + [a + (36 – 1)d] + [a + (37 – 1)d] = 429

⇒ [a + 34d] + [a + 35d] + [a + 36d] = 429

⇒ 3a + 105d = 429

Dividing by 3

⇒ a + 35d = 143 …….(2)

Subtracting eq. (1) from eq. (2) we get,

⇒ [a + 35d] – [a + 18d] = 143 – 75

⇒ 17d = 68

Substituting value of ‘d’ in eq. (1) we get,

⇒ a + 18 × 4 = 75

⇒ a + 72 = 75

⇒ a = 75 – 72 = 3

⇒ a = t_{1} = 3

We know that, t_{n + 1} = t_{n} + d

t_{2} = t_{1} + d = 3 + 4 = 7

t_{3} = t_{2} + d = 7 + 4 = 11

t_{4} = t_{3} + d = 11 + 4 = 15

t_{37} = 3 + (37 – 1) × 4

t_{37} = 3 + 36 × 4

t_{37} = 3 + 144 = 147

Thus, the A.P. is 3, 7, 11, . . . . ., 147

Rate this question :

Attempt any three sub-question from the following:

Find the first three terms to the sequence, whose nth term is t_{n} = 4n – 3

Attempt any four sub-questions from the following:

Find the first four terms in an A.P. When a = 10 and d = 3.

Maharashtra Board - Algebra PapersAttempt of the following question:

Write the first three terms of the A.P. whose common difference is -3 and first term is 4.

Maharashtra Board - Algebra PapersAttempt of the following sub questions:

Find the eighteenth term of the A.P.:1, 7, 13, 19, …

Maharashtra Board - Algebra PapersChoose the correct alternative answer for each of the following sub questions.

For an given A.P. a = 3.5, d = 0, n = 101, then t_{n} = . . .

Attempt of the following question:

Write the first two terms of the sequence whose nth term is t_{n} = 3n – 4.

Choose the correct alternative answer for each of the following sub questions.

If for any A.P. d = 5 then t_{18} – t_{13} = ...

Choose the correct alternative answer for each of the following sub questions.

In an A.P. first two terms are – 3, 4 then 21^{st} term is . . .

Attempt of the following question:

Find the sum of all numbers from 50 to 350 which are divisible by 6. Hence find the 15th term of that A.P.

Maharashtra Board - Algebra PapersAttempt any three of the following sub questions:

How many three digit natural numbers are divisible by 5?

Maharashtra Board - Algebra Papers