Let's look at the meanings of the given words.
Aimed: point or direct (a weapon or camera) at a target
Manifest: show (a quality or feeling) by one's acts or appearance; demonstrate.
Provided: make available for use; supply.
Nurtured: care for and protect (someone or something) while they are growing.
Declined: politely refuse (an invitation or offer).
The sentence talks about the help from the Western allies. The blank requires a past participle verb because it follows the helping verb 'have'. According to this, option (b) is eliminated as it is a base verb. The verbs 'aimed' and 'nurtured' don't provide coherent meanings. The subject ('He') wanted to thank the western allies. Therefore, of the remaining two options, 'provided' is the appropriate choice because he is thanking them for the help they provided.
So, the correct sentence is: He wanted to thank Western allies for the help they have provided so far.
Hence, option (c) is the correct answer.
Note:
Participle: A word formed from a verb (e.g. going, gone, being, been) and used as an adjective (e.g. working woman, burnt toast) or a noun (e.g. good breeding). In English, participles are also used to make compound verb forms (e.g. is going, has been). Consider the sentences given below:
Dancing at the party, the women fell on the ground.
Encouraged by her parents, Rita went on to pursue her passion.
The highlighted word in the first sentence is of the form Verb + ing and is used as a present participle. It is used to denote an action that is going on or incomplete or imperfect.
The highlighted word in the second sentence is of the form verb + ed. It is used as a past participle. It denotes the completed action or state of the thing spoken of.
A past participle ends with the following: -ed, -d, -t, -en, or -n.
One should take great care and never get confused between a gerund and a present participle. They use the same forms but perform different functions: a gerund is a verbal noun, while a present participle is a verbal adjective.
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