Your question involves electron configurations, valence electrons, and the periodic table, all in one. There are multiple ways to solve this question, all arriving at the same answer. I'm not sure which makes the most sense to you, but here it goes:
1. The Aufbau Principle (long way)
For this question, you no doubt have already referenced and learned the Aufbau principle for electrons, the order at which electrons fill the various energy levels and sublevels around a nucleus. While known the order is best, don't be afraid to look it up in your textbook. You'll find a variety of ways describing it, but you should see the diagonal rule attached to this answer which guides you through them. You should know about the different sublevels (s,p,d, and f) and the number of electrons each can hold (2, 6, 10, and 14). When looking at the coefficients of each letter, you're actually seeing the different energy levels that the sublevels belong to. For your question, you want the element that has three energy levels and 7 valence electrons, or 7 electrons on the third energy level. If you trace through the image, the first sublevel you hit on the third energy level is the 3s, which can only hold 2 electrons. Since you want 7 valence electrons, you have to account for the other 5. If you keep tracing the image, you then come to the 3p sublevel, which can hold 6, but you only have 5 left. This means that the last part of your configuration should be 3p^5. According to the Aufbau principle, each energy level (and sublevel) should be filled before you fill the next. In other words, everything before 3p should be full. Trace along the diagonal rule and you arrive at:
1s^2 2s^2 2p^6 3s^2 3p^5
Each superscript (exponent) tells you how many electrons are in each sublevel. If you add them all up, you have 17. Thus, this element that has three energy levels and 7 valence electrons must have 17 electrons total. When you look at the periodic table, chlorine, Cl, is atomic number 17.
2. The Aufbau Principle and the Periodic Table (the easier way)
The more electron configurations you work out and study, the more comfortable you will become with them. The odd shape of the periodic table stems from the arrangement of all the electrons around a nucleus for each element. Attached, you'll find another image for the Aufbau principle, except it overlaps the periodic table. To make sense of it, start in the top left at hydrogen, and then trace your finger to the right (helium). Then go to row two, starting at where lithium is found and again trace right until you find neon. If you continue in this pattern, you are following the order of the sublevels. Your question asks for an element with three energy levels and 7 valence electrons. If you trace the periodic table, the first time you find something on the third energy level is 3s (the third row). If you trace that row, you'll move one spot over and fill the 3s sublevel. You would have covered 2 elements for the 2 electrons an s can hold. 5 more valence electrons have to be accounted for. If you continue tracing that row, counting along the way, you'll find yourself in the p-block. If you count to five in the p-block, you'll stop in the position that chlorine can be found on a standard periodic table. It is the 7th element on row three of the periodic table, 5th element in p-block on row three. Your answer is again chlorine, Cl.
3. The periodic table and groups (easiest way)
The group numbers on the periodic table reference electron configurations. Elements in group 17 or group VIIA (same group), they will have 7 valence electrons. The row or period an element is found on references the highest occupied energy level for that element. For an element to have three occupied energy levels and 7 valence electrons, it must be on row three and in group 17 or group VIIA. Again, you find chlorine, Cl.
They each arrive at the same answer. The only difference is which way your brain likes to see the issue.
Use the Periodic Table to answer such questions. The valence electrons are those in the outermost energy level and are often used in bonding. Each family (or group or column) on the PT have a specific number of valence electrons. For example Group 1, the alkali metals, (Li, Na, K, etc.) have 1 valence electron. Group 2, the alkaline earth metals, (Be, Mg, Ca, etc.) have 2 valence electrons. Groups 3-12, the transition metals, have 2 valence electrons (most of them) but some have d electrons that may be involved in bonding too. Group 13 elements have 3 valence electrons. Group 14 elements have 4 valence electrons. Group 15 elements have 5 valence electrons. Group 16 elements have 6 valence electrons. Group 17 elements have 7 valence electrons. Group 18 elements have 8 valence electrons.
So from the description, you can conclude that the element described, with 7 valence electrons, must be in group 17.
Next you need to decipher what the three energy levels indicates. The electrons fill up the energy levels in the electron cloud from the lowest level first. When that energy level is filled, the electrons begin to fill the next highest energy level. Interestingly, the periodic table is set up such that elements in the first row (Period 1) have their valence electrons in the first energy level. Period 2 elements have their valence electrons in the second energy level, etc.
So from the description, the element has three energy levels. It's valence electrons must be in the third energy level, and therefore the element must be in Period 3 on the Periodic Table.
Thus the element described must be in Period 3, Group 17, which is Chlorine, symbol Cl.
What is the the SYMBOL of the element that has three energy levels and seven valence electrons?
This can be easily done using your periodic table. The rows of a periodic table indicate the number of energy levels an element has. So to find an element with three energy levels you want to go to the third row down. Now to find the element with seven valence electrons you can use the columns (aka groups) an element is in. The first column means those elements have 1 valence electron, the second column means those elements have 2 valence electrons, and so on. The only tricky part is you have to skip the transition metals (the middle group). The trend picks up again on column 13, those elements have 3 valence electrons, group 14 has 4 valence electrons, group 15 has 5 valence electrons, group 16 has 6 valence electrons, and group 17 has 7 valence electrons. So we are looking for the element in the 3rd row and 17th group or column and there we will find Chlorine, symbol Cl.
This is a very broad science question regarding elements. Based on the information that was provided one can narrow down the pool of elements to a specific family or group of elements.
The three orbitals is important information but it is general because most elements that have over 11 electrons will have a third orbital. The significant bit of information provided was the valence electrons. Valence electrons tells us that it has seven electrons in its outermost orbital. With that in mind, we can narrow the element down to the Halogens or group 17.
In terms of which specific element, the clue that the valence electrons are in the third orbital narrows it to Chlorine. The remaining elements would not have seven valence electrons in the third orbital.
The chemical element that you are asking about is chlorine. Its periodic table symbol, or abbreviation, is Cl. It has 17 total electrons that orbit the nucleus. The first energy level can hold 2 electrons. That leaves 15 electrons left over. The second energy level can hold 8 electrons, which leaves 7 electrons to fill the third energy level. They are the final 7 electrons in the outermost energy level, so they are referred to as valence electrons. If you are looking at a periodic table, any element in the same column as chlorine also has 7 valence electrons. Flourine has 7 on the second level, chlorine has 7 on the third level, bromine has 7 on the fourth level, etc.
The answer is Cl.
Group/family 18, the Nobel Gases, have full octets of 8 valence electrons. Each time you move to the LEFT one time within the nonmetals on the Periodic Table, an electron is lost. So, any element in group/family 17 would have 7 valence electrons (you can check this by drawing Bohr diagrams for each element within group/family 17).
So, we now know that the element is a nonmetal found in group/family 17. Therefore, it is found in the "p" block. For elements found in the "p" block, periods represent energy levels.
Thus, if I match the third period with group/family 17, I have found my answer- Chlorine!