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

We have, T_n = (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

Let 1^st 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

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

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

We have given that a_p = q and a_q = 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
∴ a_n = a + (n – 1) d = q + p – 1 + (n – 1) (– 1)
= q + p – 1 + 1 – n = q + p – n

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
∴ 8^th term = a₁ + 7d = 1 + 14 = 15

We have, given that, 

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

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

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

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, 

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.