Math Olympiad Test: Arithmetic Progression- 3 - Class 10 MCQ

# Math Olympiad Test: Arithmetic Progression- 3 - Class 10 MCQ

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## 10 Questions MCQ Test - Math Olympiad Test: Arithmetic Progression- 3

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Math Olympiad Test: Arithmetic Progression- 3 - Question 1

### Find the sum of first 20 terms of an A.P. whose nth term is given by Tn = (7 – 3n).

Detailed Solution for Math Olympiad Test: Arithmetic Progression- 3 - Question 1

We have, Tn = (7 – 3n)
First term, T1 = (7 – 3 × 1) = 4
Second term, T2 = 7 – 3 × 2 = 1
Third term, T3 = 7 – 3 × 3 = –2
∴ Series is 4, 1, – 2, .......... and common difference = –3
Sum of first 20 terms (S20)
= 20/2 [24 (20 - 1)(- 3)] = 10 [8 – 57] = –490

Math Olympiad Test: Arithmetic Progression- 3 - Question 2

### If 9th term of an A.P. is zero, then its 29th term is ________ its 19th term.

Detailed Solution for Math Olympiad Test: Arithmetic Progression- 3 - Question 2

Let 1st term of A.P. be a and common difference be d.
Now, a9 = 0 ⇒ a + 8d = 0 ⇒ a = –8d ...(i)
Now, a29 = a + 28d = –8d + 28d
⇒ a29 = 20d ...(ii)
Also, a19 = a + 18d = –8d + 18d = 10d
⇒ 2 × a19 = 2 × 10d = 20d ...(iii)
From (ii) and (iii), we have
a29 = 2a19

Math Olympiad Test: Arithmetic Progression- 3 - Question 3

### Which term of the A.P. 5, 2, –1, ....... is –22?

Detailed Solution for Math Olympiad Test: Arithmetic Progression- 3 - Question 3

Given A.P. is 5, 2, –1, ......
⇒ a = 5, d = 2 – 5 = –3
Tn = – 22 ⇒ a + (n – 1)d = –22
⇒ 5 + (n – 1) (–3) = –22 ⇒ n = 10
Hence, 10th term of the given A.P. is –22.

Math Olympiad Test: Arithmetic Progression- 3 - Question 4

In an A. P., the sum of first n terms is  Find its 25th term.

Detailed Solution for Math Olympiad Test: Arithmetic Progression- 3 - Question 4

We have given that
25th term = Sum of 25 terms – Sum of 24 terms
= S25 – S24
Now, S25 = 1100 and S24 = 1020
∴ 25th term = 1100 – 1020 = 80

Math Olympiad Test: Arithmetic Progression- 3 - Question 5

In an A.P., if the pth term is ‘q’ and the qth term is ‘p’, then its nth term is ________.

Detailed Solution for Math Olympiad Test: Arithmetic Progression- 3 - Question 5

We have given that ap = q and aq = p
⇒ q = a + (p – 1)d and ... (i)
p = a + (q – 1)d ... (ii)
Subtracting (ii) from (i), we get
q – p = d (p – q) ⇒ d = –1
Now, q = a + 1 – p [From (i)]
⇒ a = q + p – 1
∴ an = a + (n – 1) d = q + p – 1 + (n – 1) (– 1)
= q + p – 1 + 1 – n = q + p – n

Math Olympiad Test: Arithmetic Progression- 3 - Question 6

The sum of all terms of the arithmetic progression having ten terms except for the first term, is 99, and except for the sixth term, is 89. Find the 8th term of the progression if the sum of the first and the fifth term is equal to 10.

Detailed Solution for Math Olympiad Test: Arithmetic Progression- 3 - Question 6

According to the question, we have
a2 + ....... + a10 = 99 ... (i)
and a1 + ....... + a5 + a7 +..... + a10 = 89 ... (ii)
Subtracting (ii) from (i), we get
⇒ a6 – a1 = 10 ⇒ a1 + 5d – a1 = 10
⇒ 5d = 10 ⇒ d = 2
Also, a1 + a5 = 10 ⇒ a1 + a1 + 4d = 10
⇒ 2a1 + 8 = 10 ⇒ a1 = 1
∴ 8th term = a1 + 7d = 1 + 14 = 15

Math Olympiad Test: Arithmetic Progression- 3 - Question 7

The ratio of the sum of m and n terms of an A.P. is m2 : n2, then find the ratio of mth and nth terms.

Detailed Solution for Math Olympiad Test: Arithmetic Progression- 3 - Question 7

We have, given that,

Replacing m with 2m – 1 and n with 2n – 1, we get

Math Olympiad Test: Arithmetic Progression- 3 - Question 8

If x ≠ y and the sequences x, a1, a2, y and x, b1, b2, y each are in A.P., then  is ________.

Detailed Solution for Math Olympiad Test: Arithmetic Progression- 3 - Question 8

For sequence, x, a1, a2, y
y = x + 3d ⇒ d = (y - x)/3

Similarly,
For sequence, x, b1, b2, y

Math Olympiad Test: Arithmetic Progression- 3 - Question 9

If the mth term of an A.P. is 1/n and nth term is 1/m, then the sum of first mn terms is _______.

Detailed Solution for Math Olympiad Test: Arithmetic Progression- 3 - Question 9

We have given,
Then, a + (m-1)d = 1/n ...(i)
And, a + (n-1)d = 1/m ... (ii)
Subtracting (ii) from (i), we get,

[From (i)]

Now,

Math Olympiad Test: Arithmetic Progression- 3 - Question 10

Four numbers are inserted between the numbers 4 and 39 such that an A.P. results. Find the biggest of these four numbers.

Detailed Solution for Math Olympiad Test: Arithmetic Progression- 3 - Question 10

Since 4 terms are inserted between 4 and 39 so there are 6 terms in the AP
Therefore, the terms of the AP ar given by: 4, 4 + d, 4 + 2d, 4 + 3d, 4 + 4d, 4+5d
⇒ 39 = 4 + 5d,
⇒ d = 7,
So, as it is an increasing AP, therefore 4+4d would be the biggest.
⇒ 4 + 4d = 4 + 4 × 7 = 4 + 28 = 32.