The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by . Then the number of terms which are integers in the A.P. is : [2025]

Question

The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$ . Then the number of terms which are integers in the A.P. is : [2025]

Smart Achievers · Accepted Answer

(2)

Let n be the number of terms of an A.P., $a$ be the first term and $d$ be the common difference.

Let n = 2m, so terms are a, a + d, a + 2d, a + (2m – 1)d

Odd terms = a, a + 2d, a + 4d, ...., a + (2m – 2)d

Even terms = a + d, a + 3d, a + 5d, .... a + (2m – 1)d

Now, $S_{even} - S_{odd}$ = (a + d – a) + (a + 3d – a – 2d) + .... + (a + (2m – 1)d – a – (2m – 2)d)

$\Rightarrow 30 - 24 = m d \Rightarrow m d = 6$ ... (i)

Now, $a + (2 m - 1) d - a = \frac{21}{2} \Rightarrow 2 m d - d = \frac{21}{2}$

$\Rightarrow 12 - d = \frac{21}{2} \Rightarrow d = 12 - \frac{21}{2} = \frac{3}{2}$ [From (i)]

Since, $m d = 6 \Rightarrow m \times \frac{3}{2} = 6 \Rightarrow m = 4$

Now, $n = 2 m = 2 \times 4 = 8$

So, number of terms = 8.

Solution

Q.
The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$ . Then the number of terms which are integers in the A.P. is : [2025]

1 10

2 8

3 4

4 6

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Solution

Q. The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by 212. Then the number of terms which are integers in the A.P. is : [2025]

1 10

2 8

3 4

4 6

Want Us to Email you About Special Offers & Updates?

Q.
The number of terms of an A.P. is even; the sum of all the odd terms is 24, the sum of all the even terms is 30 and the last term exceeds the first by $\frac{21}{2}$ . Then the number of terms which are integers in the A.P. is : [2025]