Practice Questions for Algebra - Progressions

1. PE 1 - Progressions | Algebra - Progressions

The sum to 50 terms of a series in arithmetic progression is 1275 and the 5^th term is 5. What are the values of the first term and the common difference respectively?

(a)
2, 2
(b)
2, 1
(c)
1, 2
(d)
1, 1

2. PE 1 - Progressions | Algebra - Progressions

The sum of the terms of an arithmetic progression of 50 terms is 2500. The sum of the least 10 terms of the progression is 100. Find the largest term of the progression.

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3. PE 1 - Progressions | Algebra - Progressions

Find the sum of all the perfect squares between 100 and 1000 (excluding 100 and 1000).

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Answer: 10031

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Explanation :

We know that the sum of perfect squares of first n natural numbers i.e.,

1² + 2² + 3² + … + n² = $\frac{n (n + 1) (2 n + 1)}{6}$

Perfect square just less than 1000 is 31² = 961.
Perfect square just greater than 100 is 11² = 121.

The required sum = 11²+ 12² + 13² + … + 31²

∴ Sum of perfect square till 1000 i.e.,

1²+ 2² + 3² + … + 31² = $\frac{31 (31 + 1) (2 \times 31 + 1)}{6}$ = 10416

Also, sum of perfect square till 100 i.e.,

1² + 2² + 3² + … + 10² = $\frac{10 (10 + 1) (2 \times 10 + 1)}{6}$ = 385

∴ Sum of perfect square between 100 and 1000 = 10416 – 385 = 10031.

Hence, 10031.

4. PE 1 - Progressions | Algebra - Progressions

Find the sum of all the numbers between 99 and 999 which when divided by 7 leaves a remainder of 4.

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5. PE 1 - Progressions | Algebra - Progressions

Find the number of terms, in the series 16, 14, 12, … such that the sum is 66.

(a)
6
(b)
11
(c)
Either (a) or (b)
(d)
Neither (a) nor (b)

6. PE 1 - Progressions | Algebra - Progressions

Find the maximum value of n for which the sum to n terms of the progression 9, 13, 17, ….. is less than 600.

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7. PE 1 - Progressions | Algebra - Progressions

Find the sum of the squares of the first 10 terms of the series whose sum up to n terms is given by n² + 2n.

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Answer: 1770

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Explanation :

S_n = n² + 2n

T_n = S_n – S_n-1

∴ T_n = (n² + 2n) - ((n – 1)² + 2(n – 1)) = 2n + 1

Sum of square of n terms = Σ(2n + 1)²

= Σ(4n² + 2n + 1)

= 4Σn² + 4Σn + Σ1

= 4 $(\frac{n (n + 1) (2 n + 1)}{6})$ + 4 $(\frac{n (n + 1)}{2})$ + n

∴ Sum of square of 10 terms = 4 $(\frac{10 (10 + 1) (20 + 1)}{6})$ + 4 $(\frac{10 (10 + 1)}{2})$ + 10

= 1540 + 220 + 10 = 1770.

Hence, 1770.

8. PE 1 - Progressions | Algebra - Progressions

If $\log_{x} 2$ + $\log_{x^{1 / 2}} 2$ + $\log_{x^{1 / 4}} 2$ + $\log_{x^{1 / 8}} 2$ + ... upto 10 terms is 1023, find x.

(a)
2
(b)
4
(c)
16
(d)
8

Answer: Option A

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Explanation :

If $\log_{x} 2$ + $\log_{x^{1 / 2}} 2$ + $\log_{x^{1 / 4}} 2$ + $\log_{x^{1 / 8}} 2$ + ... upto 10 terms is 1023.

⇒ log_x2 + 2 log_x2 + 2² log_x2 + … + 2⁹ log_x2 = 1023

⇒ (1 + 2 + 2² + … + 2⁹) log_x2 = 1023

⇒ (2¹⁰ - 1)log_x2 = 1023

⇒ (1024 - 1) × log_x2 = 1023

⇒ 1023 × log_x2 = 1023

⇒ log_x2 = 1

⇒ x = 2

Hence, option (a).

9. PE 1 - Progressions | Algebra - Progressions

The sum of the first 23 terms of an arithmetic progression equals the sum of its first 34 terms. Find which term of this series is 0.

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10. PE 1 - Progressions | Algebra - Progressions

The sum of the first 23 terms of an arithmetic progression equals the sum of its first 34 terms. Find the sum of its first 57 terms.

(a)
0
(b)
-1
(c)
1
(d)
Cannot be determined

PE 1 - Progressions | Algebra - Progressions

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