Question

Write out the following sums, one term for each value of k. Simplify each term as much as possible, but do not enter decimals. For example, enter 1+4+9 instead of 12+22+32 or 14, or enter 1/2+1/2 instead of 0.5+0.5 or 1. The purpose of this problem is for you to show that you know how to interpret summation notation and write all of the terms in a sum, which is why you are being told not to reduce your answers very much.

Answer 1

The correct question is:

Write out the following sums, one term for each value of k. Simplify each term as much as possible, but do not enter decimals. For example, enter 1 + 4 + 9 instead of 1² + 2² + 3² or 14, or enter 1/2 + 1/2 instead of 0.5 + 0.5 or 1.

The purpose of this problem is for you to show that you know how to interpret summation notation and write all of the terms in a sum, which is why you are being told not to reduce your answers very much.

`(a) \sum_(k=0)^5 2^k \\ \\(b) \sum_(k=2)^7 (1)/(k) \\ \\(c) \sum_(k=1)^5 k^2 \\ \\(d) \sum_(k=1)^6 (1)/(6) \\ \\(e) \sum_(k=1)^6 2k`

Answer:

`(a) \sum_(k=0)^5 2^k = $1 + 2 + 4 + 8 + 16 + 32$ \\ \\(b) \sum_(k=2)^7 (1)/(k) = (1)/(2) + (1)/(3) + (1)/(4)+ (1)/(5)+ (1)/(6)+ (1)/(7) \\ \\(c) \sum_(k=1)^5 k^2 = 1 + 4 + 9 + 16 + 25 \\ \\(d) \sum_(k=1)^6 (1)/(6) = (1)/(6) + (1)/(6) + (1)/(6) + (1)/(6) + (1)/(6) + (1)/(6) \\ \\(e) \sum_(k=1)^6 2k = 2 +4 +6 +8 +10 +12`

Explanation:

`(a) \sum_(k=0)^5 2^k\\For k = 0: 2^k = 2^0 = 1\\For k = 1: 2^1 = 2\\For k = 2: 2^2 = 4\\For k = 3: 2^3 = 8\\For k = 4: 2^4 = 16\\For k = 5: 2^5 = 32\\\sum_(k=0)^5 2^k = 1 + 2 + 4 + 8 + 16 + 32`

`(b) \sum_(k=2)^7 (1)/(k)\\For k = 2: 1/2\\For k = 3: 1/3\\For k = 4: 1/4\\For k = 5: 1/5\\For k = 6: 1/6\\For k = 7: 1/7\\ \sum_(k=2)^7 (1)/(k) = 1/2 + 1/3 + 1/4 + 1/5 + 1/6+ 1/7`

`(c) \sum_(k=1)^5 k^2\\For k = 1: 1^2 = 1\\For k = 2: 2^2 = 4\\For k = 3: 3^2 = 9\\For k = 4: 4^2= 16\\For k = 5: 5^2 = 25\\\sum_(k=1)^5 k^2 = 1 + 4 + 9 + 16 + 25`

`(d) \sum_(k=1)^6 (1)/(6)\\For k = 1: 1/6\\For k = 2: 1/6\\For k = 3: 1/6\\For k = 4: 1/6\\For k = 5: 1/6\\For k = 6: 1/6\\ \sum_(k=1)^6 (1)/(6) = 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6`

`(e) \sum_(k=1)^6 2k\\For k = 1: 2*1 = 2\\For k = 2: 2*2 = 4\\For k = 3: 2*3 = 6\\For k = 4: 2*4 = 8\\For k = 5: 2*5 = 10\\For k = 6: 2*6 = 12\\\sum_(k=1)^6 2k = 2 +4 +6 +8 +10 +12`