الرقم التسلسلي في الصورة.

A careful MSA reading is:

ar-raqmu at-tasalsuliyyu fī aṣ-ṣūrati

If case endings are not pronounced, learners often hear something closer to:

ar-raqm at-tasalsulī fī aṣ-ṣūra

A few pronunciation notes:

• الرقم is pronounced ar-raqm, not al-raqm
• التسلسلي is pronounced at-tasalsulī, not al-tasalsulī
• الصورة is pronounced aṣ-ṣūra, not al-ṣūra

That happens because ر, ت, and ص are sun letters, so the l sound of ال assimilates to the next consonant.

It breaks down like this:

• الرقم = the number
• التسلسلي = serial / sequential
  Together, الرقم التسلسلي = the serial number
• في = in
• الصورة = the picture / image

So structurally it is:

• a noun phrase: الرقم التسلسلي
• followed by a prepositional phrase: في الصورة

Because in Arabic, adjectives normally come after the noun they describe.

So Arabic says:

• الرقم التسلسلي
• literally: the number serial

But natural English reverses that order:

• the serial number

This is one of the most common word-order differences between Arabic and English.

Because Arabic adjectives must agree with the noun they describe in definiteness.

Here:

• الرقم is definite because it has ال
• so the adjective التسلسلي must also be definite

That is why you get:

• الرقم التسلسلي = the serial number

If the noun were indefinite, the adjective would also be indefinite.

Yes. It is an adjective meaning something like serial, sequential, or related to sequence.

It behaves like a normal Arabic adjective, so it agrees with the noun in:

• definiteness
• gender
• number
• case

Here it matches الرقم, which is:

• masculine
• singular
• definite

So the adjective appears in the matching form: التسلسلي.

It can be understood either way, depending on context.

As written, it can be read as a noun phrase + prepositional phrase:

• الرقم التسلسلي في الصورة
• the serial number in the picture

But in context, Arabic can also use this kind of structure as a verbless present-tense sentence, meaning:

• The serial number is in the picture

Arabic often leaves out is / are in the present tense.

In Arabic, present-tense nominal sentences usually do not use a separate word for is.

So instead of saying something equivalent to:

• The serial number is in the picture

Arabic simply says:

• الرقم التسلسلي في الصورة

The meaning of is is understood from the structure.

في is a preposition meaning in.

After a preposition, the following noun goes into the genitive case in MSA. So:

• في = in
• الصورةِ = the picture / image in the genitive

In a fully vowelled version, you would write:

فِي الصُّورَةِ

The final -i sound shows the genitive case, although in normal unvowelled writing that ending is usually not shown.

A fully vowelled version would be:

الرَّقْمُ التَّسَلْسُلِيُّ فِي الصُّورَةِ

Why those endings?

• الرَّقْمُ is nominative
• التَّسَلْسُلِيُّ matches it as its adjective, so it is also nominative
• الصُّورَةِ is genitive because it comes after في

In everyday writing, these short case endings are usually omitted, so you normally see:

الرقم التسلسلي في الصورة