The image shows a mathematics problem set, likely related to arithmetic sequences and the sum of integers. Here's a breakdown of what I can see and analyze:

1. Problem (a):

   • The first problem seems to be about finding the sum of the series $2 + 5 + 8 + 11 + \dots$ up to the 12th term.
   • This is an arithmetic series where:
     • The first term $a_1 = 2$,
     • The common difference $d = 3$,
     • The number of terms $n = 12$.

   To find the sum $S_n$ of the first $n$ terms of an arithmetic series, you can use the formula: $S_n = \frac{n}{2} \times (2a_1 + (n - 1)d)$ Plugging in the values:

   = 6 \times (4 + 33) = 6 \times 37 = 222$$ So, the sum of the first 12 terms is **222**.

2. Problem (b):

   • This involves calculating the sum of all the odd integers between 50 and 200.

   Odd numbers between 50 and 200 start at 51 and end at 199. This is also an arithmetic sequence where:

   • The first term $a_1 = 51$,
   • The common difference $d = 2$,
   • The last term $l = 199$.

   To find the number of terms $n$, use: $n = \frac{l - a_1}{d} + 1 = \frac{199 - 51}{2} + 1 = \frac{148}{2} + 1 = 74 + 1 = 75$

   Now, use the sum formula: $S_n = \frac{n}{2} \times (a_1 + l) = \frac{75}{2} \times (51 + 199) = 37.5 \times 250 = 9375$ So, the sum is 9375.

3. Problem (c):

   • This one asks for the sum of all the multiples of 7 between 100 and 200.

   Multiples of 7 between 100 and 200 start at 105 and end at 196. Again, this is an arithmetic sequence where:

   • The first term $a_1 = 105$,
   • The common difference $d = 7$,
   • The last term $l = 196$.

   Find the number of terms $n$: $n = \frac{l - a_1}{d} + 1 = \frac{196 - 105}{7} + 1 = \frac{91}{7} + 1 = 13 + 1 = 14$

   Use the sum formula: $S_n = \frac{n}{2} \times (a_1 + l) = \frac{14}{2} \times (105 + 196) = 7 \times 301 = 2107$ So, the sum is 2107.

Do you want further details or have any questions?

5 Related Questions:

1. How do you find the nth term of an arithmetic sequence?
2. What is the difference between arithmetic and geometric sequences?
3. How can you calculate the sum of even integers between two numbers?
4. What is the formula for the sum of an infinite geometric series?
5. How does the common difference affect the sum of an arithmetic sequence?

Tip:

When dealing with arithmetic sequences, always identify the first term and the common difference. This simplifies the process of finding the number of terms and calculating the sum.