Worksheet: Arithmetic Series

In this worksheet, we will practice calculating the sum of the terms in an arithmetic sequence with a definite number of terms.

Q1:

Find the number of terms in the arithmetic sequence whose first term is 11 and last term is 81, where the sum of all the terms is 506.

Q2:

Find the number of terms for which the sum of the sequence (160,144,128,) is 0.

Q3:

The house numbers on one side of a street are 1,2,3,4,…,49. Find the house number where the sum of the numbers before it is equal to the sum of the numbers after it.

  • A35
  • B34
  • C33
  • D36
  • E37

Q4:

Find, in terms of 𝑛, the sum of the arithmetic sequence (9,10,11,…,𝑛+8).

  • A𝑛2(𝑛+17)
  • B𝑛2(𝑛+8)
  • C𝑛2(𝑛+9)
  • D𝑛2(𝑛+10)
  • E𝑛(𝑛+17)

Q5:

Find the sum of the 6 consecutive terms that start from the eighteenth term of the arithmetic series 16+23+30+⋯.

Q6:

Find the least number of terms needed to make the sum of the arithmetic sequence (15,10,5,…) negative.

Q7:

Find the sum of the sequence of odd natural numbers which are greater than 46 and less than 92.

Q8:

Find the 12-term arithmetic sequence given the sum of the first 4 terms is −158 and the sum of the last 4 terms is −574.

  • A(−20,−59,−124,…)
  • B(−163,−150,−137,…)
  • C(−13,−33,−53,…)
  • D(−20,−33,−46,…)
  • E(−20,−7,6,…)

Q9:

Find the arithmetic sequence given the sum starting from the second term is 567, the sum excluding the last term is 469 and the difference between the tenth and sixth terms is 28.

  • A(−161,−168,−175,…,−5)
  • B(7,−5,−17,…,−161)
  • C(79,72,65,…,−12)
  • D(−12,−19,−26,…,−110)
  • E(−12,−5,2,…,86)

Q10:

Find the arithmetic sequence that starts at 𝑎 given that the sum of the first 17 terms is 748 and the arithmetic mean between 𝑎 and 𝑎 is 48.

  • A(76,72,68,…)
  • B(76,80,84,…)
  • C(−4,72,148,…)
  • D(−76,−72,−68,…)
  • E(−76,−80,−84,…)

Q11:

Find all possible arithmetic sequences that satisfy 𝑆=0 and 𝑎×𝑎=−216 where 𝑆 is the sum of the first 𝑛 terms of the arithmetic sequence, 𝑎 is the 𝑛th term, and 𝑎 is the first term.

  • A(6,−60,−126,…) or (−6,60,126,…)
  • B(66,72,78,…) or (−66,−72,−78,…)
  • C(−66,−72,−78,…) or (66,72,78,…)
  • D(−66,−60,−54,…) or (66,60,54,…)

Q12:

In an arithmetic sequence that starts at 𝑎, 𝑎=29 and the sum of the eighth and the twelfth terms exceeds the fifteenth by 7. Find the number of terms whose sum is 322.

Q13:

Find the sum of a sequence which has 9 terms, given that the sum of the first third of the terms is 102 and that of the last third of the terms is −186. Then, find the sum of the middle five terms.

  • A−252,−56
  • B−112,−230
  • C−126,−70
  • D−140,10
  • E−84,162

Q14:

The 𝑛th term of an arithmetic sequence is 𝑎 and the first term is 𝑎.

Find the sum of the positive terms of the arithmetic sequence in which 𝑎=−54 and 𝑎=−84.

Q15:

Find all possible values of 𝑛 for which the sum of the sequence 64,56,48,… is 264.

  • A7 or 12
  • B11 or 6
  • C8 or 13
  • D10 or 5
  • E9 or 4

Q16:

The 𝑛th term of an arithmetic sequence is 𝑎 and the first term is 𝑎.

Find the sum of all the terms from 𝑎 to 𝑎 for the arithmetic sequence that starts 4+8+12+⋯.

Q17:

You open a book to the middle and see page numbers 12 and 13. What is the sum of all the page numbers in the book?

Q18:

Find 𝑛 given that the sum of the first 𝑛 terms of the arithmetic sequence (−18,−21,−24,…) is 21 times greater than the sum of the first 𝑛 terms of the arithmetic sequence (−69,−54,−39,…).

Q19:

Find the sum of the first 16 terms of the sequence 𝑎=11𝑛−41,𝑛,8𝑛+49,𝑛.isoddiseven

Q20:

The 𝑛th term of an arithmetic sequence is 𝑎 and the first term is 𝑎.

Find 𝑎 given that the sum of the first 𝑛 terms of the arithmetic sequence is 𝑆=𝑛2(−9𝑛+105).

Q21:

Find the sum of the sequence of natural numbers which are greater than 64, less than 74, and divisible by 3.

Q22:

The 𝑛th term of an arithmetic sequence is 𝑎, the first term is 𝑎, and the sum of the first 𝑛 terms is 𝑆.

Find 𝑆 given that 𝑆−𝑆=−52.

Q23:

5 arithmetic means were inserted between 𝑎 and 𝑏, and another 3 arithmetic means were inserted between the same two numbers. If the sum of the 5 means exceeds that of the 3 by 112 and the first mean of the 5 exceeds that of the 3 by 2, find 𝑎 and 𝑏.

  • A𝑎=62, 𝑏=50
  • B𝑎=72, 𝑏=40
  • C𝑎=74, 𝑏=38
  • D𝑎=64, 𝑏=48
  • E𝑎=68, 𝑏=44

Q24:

Find the sum of the first 35 terms in the sequence 1,16,64,256,…logloglog.

Q25:

Find the sum of the first 21 terms of the arithmetic sequence given 𝑎+𝑎=−232 and 𝑎=−130.

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