I found this post from John Piper on the Desiring God Blog very helpful.
Promises of Answered Prayer to Encourage Us to Pray with Hope
For I know the plans I have for you, declares the LORD, plans for wholeness and not for evil, to give you a future and a hope. 12 Then you will call upon me and come and pray to me, and I will hear you. 13 You will seek me and find me. When you seek me with all your heart,
2 Chronicles 7:14
If my people who are called by my name humble themselves, and pray and seek my face and turn from their wicked ways, then I will hear from heaven and will forgive their sin and heal their land.
Ask, and it will be given to you; seek, and you will find; knock, and it will be opened to you. 8 For everyone who asks receives, and the one who seeks finds, and to the one who knocks it will be opened. 9 Or which one of you, if his son asks him for bread, will give him a stone? 10 Or if he asks for a fish, will give him a serpent? 11 If you then, who are evil, know how to give good gifts to your children, how much more will your Father who is in heaven give good things to those who ask him!
Likewise the Spirit helps us in our weakness. For we do not know what to pray for as we ought, but the Spirit himself intercedes for us with groanings too deep for words.
From of old no one has heard or perceived by the ear, no eye has seen a God besides you, who acts for those who wait for him.
Before they call I will answer; while they are yet speaking I will hear.
Call to me and I will answer you, and will tell you great and hidden things that you have not known.
The LORD is good to those who wait for him, to the soul who seeks him.
The LORD is near to all who call on him, to all who call on him in truth. 19 He fulfills the desire of those who fear him; he also hears their cry and saves them.
O LORD, you hear the desire of the afflicted; you will strengthen their heart; you will incline your ear
The eyes of the LORD are toward the righteous and his ears toward their cry. . . 17 When the righteous cry for help, the LORD hears and delivers them out of all their troubles.
Again I say to you, if two of you agree on earth about anything they ask, it will be done for them by my Father in heaven. 20 For where two or three are gathered in my name, there am I among them.”
Draw near to God, and he will draw near to you.
Until now you have asked nothing in my name. Ask, and you will receive, that your joy may be full.
Delight yourself in the LORD, and he will give you the desires of your heart. 5 Commit your way to the LORD; trust in him, and he will act.
But for you, O LORD, do I wait; it is you, O Lord my God, who will answer.
He regards the prayer of the destitute and does not despise their prayer.
Whatever you ask in my name, this I will do, that the Father may be glorified in the Son. 14 If you ask me anything in my name, I will do it.
If you abide in me, and my words abide in you, ask whatever you wish, and it will be done for you.
You did not choose me, but I chose you and appointed you that you should go and bear fruit and that your fruit should abide, so that whatever you ask the Father in my name, he may give it to you.
And whatever you ask in prayer, you will receive, if you have faith.”
But I call to God, and the LORD will save me. 17 Evening and morning and at noon I utter my complaint and moan, and he hears my voice.
Then my enemies will turn back in the day when I call. This I know, that God is for me.
Call upon me in the day of trouble; I will deliver you, and you shall glorify me.
I am the LORD your God, who brought you up out of the land of Egypt. Open your mouth wide, and I will fill it.
Trust in the LORD with all your heart, and do not lean on your own understanding. 6 In all your ways acknowledge him, and he will make straight your paths.
1 John 5:14-15
And this is the confidence that we have toward him, that if we ask anything according to his will he hears us. 15 And if we know that he hears us in whatever we ask, we know that we have the requests that we have asked of him.
Do not be anxious about anything, but in everything by prayer and supplication with thanksgiving let your requests be made known to God. And the peace of God, which surpasses all understanding, will guard your hearts and your minds in Christ Jesus.
And I will put this third into the fire, and refine them as one refines silver, and test them as gold is tested. They will call upon my name, and I will answer them. I will say, ‘They are my people’; and they will say, ‘The LORD is my God.'”